DETERMINISTIC AND RELIABILITY BASED OPTIMIZATION OF INTEGRATED THERMAL PROTECTION SYSTEM COMPOSITE PANEL USING ADAPTIVE SAMPLING TECHNIQUES

Transcription

1 DETERMINISTIC AND RELIABILITY BASED OPTIMIZATION OF INTEGRATED THERMAL PROTECTION SYSTEM COMPOSITE PANEL USING ADAPTIVE SAMPLING TECHNIQUES By BHARANI RAVISHANKAR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012

2 c 2012 Bharani Ravishankar 2

3 To my parents, Ravishankar and Parameswari and my sister, Vidhya 3

4 ACKNOWLEDGMENTS First and foremost I would like to thank my advisors Dr. Bhavani Sankar and Dr. Raphael Haftka at the Department of Mechanical and Aerospace Engineering. All of your insights, guidance, and patience has been greatly appreciated. I would also like to thank the members of my advisory committee, Dr. Ashok Kumar at the Department of Mechanical and Aerospace Engineering and Dr. Gary Consolazio at the Department of Civil and Coastal Engineering. I would like thank my friends at the Center for Advanced Composites lab, Anurag, Prasanna, Min Song, Marlana, Tim and Sayan. I would also like to thank the members of the Structural and Multidiscplinary Group, Ben, Felipe, Diane, Anirban and Taiki for all their help, valuable inputs and suggestions. I would also like to thank my friends Anirudh, Sriram and Naren for all the help and support through out my PhD program, making this experience memorable. I would like to thank my family for their support and patience. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Outline of Dissertation Motivation and Objectives Thesis Organization BACKGROUND STUDY AND LITERATURE REVIEW Orbiter Thermal Protection System-30 years Legacy Tile TPS Non - Tile TPS Finite Element based Homogenization Deterministic and Reliability based optimization Deterministic Optimization Structural Reliability Use of Surrogate Models Application of Surrogates to Improve Constraint Boundaries Adaptive Sampling Efficient Global Reliability Analysis Uncertainty Modeling Estimation of Probability of Failure Moment based Methods Sampling Methods DESIGN AND ANALYSIS OF INTEGRATED THERMAL PROTECTION SYSTEM PANEL Design of ITPS Composite Panel Key Dimensions of the ITPS structure ITPS Materials Boundary Conditions Loads Finite Element Analyses Transient Heat Transfer Analysis Stress Analysis

6 3.2.3 Buckling Analysis Summary FINITE ELEMENT BASED HOMOGENIZATION A,B,D matrices Transverse Shear Stiffness Accuracy of the Homogenization Method Deflection Comparison Stress Comparison Results and Discussion A,B,D matrices - Equivalent material vs. Equivalent structure Transverse Shear Stiffness of ITPS Accuracy of the Homogenization Method Summary MONTE CARLO SIMULATIONS - RELIABILITY ESTIMATION Crude Monte Carlo Method (CMC) Separable Monte Carlo method (SMC) Error in the Probability of Failure Estimate SMC with regrouping and separable sampling of the limit state random variables Application to Failure Analysis of Composite Laminate Results and Discussion Crude and Separable Monte Carlo Method Regrouping and separable sampling of the limit state variables for improving accuracy Summary EFFICIENT GLOBAL RELIABILITY ANALYSIS (EGRA) EGRA algorithm Illustration of EGRA Summary DETERMINISTIC AND RELIABILITY BASED OPTIMIZATION Deterministic Optimization Uncertainty Modeling Estimation of System Reliability Reliability Based Optimization Results and Discussion Deterministic Optimization Sensitivity Analysis Reliability of the Deterministic Optimum Reliability based optimization

7 7.5.5 Error in the probability of failure estimate Summary CONCLUSIONS Conclusion Future Work A THERMAL PROPERTIES OF ITPS COMPONENTS B STRENGTH PROPERTIES OF ITPS MATERIALS C DESIGN OF EXPERIMENTS ADDED BY EGRA SAMPLING TECHNIQUE REFERENCES BIOGRAPHICAL SKETCH

8 Table LIST OF TABLES page 3-1 Key Dimensions of the ITPS panel Components of ITPS, materials and fiber orientation Material properties of the components of ITPS Heat flux load steps in the transient heat transfer analysis Periodic boundary conditions for the six deformations Variation of A 44 and A 55 of ITPS panel with n Tip deflection ratio along with contribution of bending and shear deformation towards deflection Material properties and uncertainty of the random variables Empirical and bootstrapping estimates of probability of failure using separable and crude Monte Carlo with N=M=500 and n = 10,000 repetitions Standard deviation and coefficient of variation of empirical and bootstrapping p f estimates using separable and crude Monte Carlo with N=M=500 and n = 10,000 repetitions for original limit state Relative contributions of response (stresses) and capacity (strengths) towards the uncertainty in p f through bootstrapping and also compared with empirical results Standard deviation and coefficient of variation of CMC and SMC for increasing sample size of M and N= Standard deviation and coefficient of variation of CMC, SMC and SMC regrouped for increasing sample size of M and N= Failure, safety factors applied on constraints Lower and upper bounds of design variables for deterministic optimization Coefficient of variation of input random variables included in the ITPS design Nominal allowable values of capacity and coefficient of variation Optimum design variables and minimized structural mass through global and EGRA DOE for deterministic optimization Comparison between the accuracy of global surrogate and EGRA surrogate at the deterministic optimum using corresponding surrogates

9 7-7 Comparison of responses at all the deterministic optima using the 60 DOE surrogate Comparison of responses at all the deterministic optima using the 20 DOE surrogate Comparison of responses at all the deterministic optima using the EGRA updated surrogate Difference in uncertainty in responses before and after sensitivity analysis Uncertainty in response due to input uncertainty Individual probabilities of failure of the deterministic optimum Lower and upper bounds of design variables for probabilistic optimization Optimum design variables and minimized structural mass through EGRA updated DOE for reliability based optimization Individual probabilities of failure of the probabilistic optimum Individual probabilities of failure of the deterministic optimum using probabilistic optimum surrogate Error in the probabilities of failure of the deterministic optimum using bootstrapping and repetitions Error in the probabilities of failure of the probabilistic optimum using bootstrapping and repetitions A-1 Density of the ITPS Components A-2 Thermal Properties of the bottom face sheet - Graphite Epoxy A-3 Thermal Properties of the top face sheet and Wrap - SiC/SiC A-4 Thermal Conductivity of the insulation foam - AETB A-5 Specific heat of the insulation foam - AETB B-1 Density of the ITPS Components C-1 Design points acquired using EGRA to improve the accuracy of Temperature boundary C-2 Design points acquired using EGRA to improve the accuracy of stress boundary121 C-3 Design points acquired using EGRA to improve the accuracy of buckling load boundary

10 C-4 Design points acquired using EGRA to improve the accuracy of reliability index boundary

11 Figure LIST OF FIGURES page 2-1 Thermal Protection System (TPS) used in space shuttles Sinusoidal core sandwiched between laminates Integrated Thermal Protection System with corrugated core Schematic view of the ITPS panel Dimensions of the ITPS panel Boundary conditions applied on the ITPS panel Heat flux profile incident on the ITPS panel Boundary conditions for the transient heat transfer analysis Temperature variation on TFS and BFS with respect to reentry time Buckled ITPS due to thermal loads Schematic view of the ITPS unit cell (RVE) Periodic boundary condition applied on the unit cell for ϵ x = Schematic view of the unit cell deformations due to strains and curvatures D ITPS model with unit cells in x- direction D model of the ITPS with unit cells in x- direction subjected to pressure load and fixed BCs Transverse shear stiffness A 55 of orthotropic panel Variation of A 44 and A 55 of ITPS panel with n Converged A 44 and A 55 of ITPS panel with n Ratio between tip deflection of the homogenized model and the 3-D ITPS model along the x and y-direction Stress (σ 11, σ 22 ) ratio between the homogenized model and the 3D ITPS model Stress (σ 11, σ 22 ) at n=10 comparison between the 3-D ITPS model and the homogenized model Vertical deflection along x-axis of the original ITPS and homogenized model due to pressure load Illustration of crude and separable Monte Carlo Method comparisons

12 5-2 Schematic representation of bootstrapping when only response is sampled Illustration of separable sampling with unit loads Composite pressure vessel with internal pressure of 100kPa and stresses acting in a small element of the vessel Distribution of stress and strength in the 2-direction (σ 2 ) showing the probable failure region Standard Deviation of CMC, SMC and regrouped limit state SMC where N =500 (fixed) and M is varying for 10,000 repetitions True contours of the Braninhoo function Target contour approximated by the initial kriging model of the Branin Function Expected Feasibility Function evaluated using the initial kriging approximation Kriging model updated with the new design and expected feasibility function Target contour approximated by the initial kriging model of the Branin Function Target contour accurately approximated by the EGRA methodology Variation of responses wrt thickness of the wrap t W Variation of critical responses wrt thickness of the bottom face sheet t B Variation of critical responses wrt thickness of the top face sheet t T Variation of critical responses wrt height of the foam h Sensitivity analysis of maximum BFS temperature Sensitivity analysis of Wrap stress σ Sensitivity analysis of buckling load factor

13 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DETERMINISTIC AND RELIABILITY BASED OPTIMIZATION OF INTEGRATED THERMAL PROTECTION SYSTEM COMPOSITE PANEL USING ADAPTIVE SAMPLING TECHNIQUES Chair: Bhavani V. Sankar Cochair: Raphael T. Haftka Major: Mechanical Engineering By Bharani Ravishankar May 2012 Conventional space vehicles have thermal protection systems (TPS) that provide protection to an underlying structure that carries the flight loads. In an attempt to save weight, there is interest in an integrated TPS (ITPS) that combines the structural function and the TPS function. This has weight saving potential, but complicates the design of the ITPS that now has both thermal and structural failure modes. The main objectives of this dissertation was to optimally design the ITPS subjected to thermal and mechanical loads through deterministic and reliability based optimization. The optimization of the ITPS structure requires computationally expensive finite element analyses of 3D ITPS (solid) model. To reduce the computational expenses involved in the structural analysis, finite element based homogenization method was employed, homogenizing the 3D ITPS model to a 2D orthotropic plate. However it was found that homogenization was applicable only for panels that are much larger than the characteristic dimensions of the repeating unit cell in the ITPS panel. Hence a single unit cell was used for the optimization process to reduce the computational cost. Deterministic and probabilistic optimization of the ITPS panel required evaluation of failure constraints at various design points. This further demands computationally expensive finite element analyses which was replaced by efficient, low fidelity surrogate 13

14 models. In an optimization process, it is important to represent the constraints accurately to find the optimum design. Instead of building global surrogate models using large number of designs, the computational resources were directed towards target regions near constraint boundaries for accurate representation of constraints using adaptive sampling strategies. Efficient Global Reliability Analyses (EGRA) facilitates sequentially sampling of design points around the region of interest in the design space. EGRA was applied to the response surface construction of the failure constraints in the deterministic and reliability based optimization of the ITPS panel. It was shown that using adaptive sampling, the number of designs required to find the optimum were reduced drastically, while improving the accuracy. System reliability of ITPS was estimated using Monte Carlo Simulation (MCS) based method. Separable Monte Carlo method was employed that allowed separable sampling of the random variables to predict the probability of failure accurately. The reliability analysis considered uncertainties in the geometry, material properties, loading conditions of the panel and error in finite element modeling. These uncertainties further increased the computational cost of MCS techniques which was also reduced by employing surrogate models. In order to estimate the error in the probability of failure estimate, bootstrapping method was applied. This research work thus demonstrates optimization of the ITPS composite panel with multiple failure modes and large number of uncertainties using adaptive sampling techniques. 14

15 CHAPTER 1 INTRODUCTION When a space vehicle reenters the atmosphere at hypersonic speeds, the vehicle s exterior is subjected to severe aerodynamic heating and pressure. To protect the vehicle from such extreme temperatures, a Thermal Protection System (TPS), is added on to the main load bearing structure. But, the TPS is incompatible with the main structure due to the mismatch in their thermal properties. This incompatibility, along with the action of numerous other loads like aerodynamic pressure, impact loads and other in-plane inertial loads may lead to the structure s failure with catastrophic consequences. Moreover, the TPS encompasses a significant portion of the vehicle s exterior constituting major part of the launch weight. Thus, it is imperative that apart from making the TPS suitable for protection purposes, it should be light weight in order to increase payloads that can be carried aboard. To accommodate these requirements, a multifunctional Integrated Thermal Protection System (ITPS) could be designed, in which the load bearing structure and the TPS are integrated into a composite sandwich panel. The greatest challenge in such a design would be in combining the various conflicting requirements of the TPS and the load bearing member into a single structure. The structural requirements favor using metals, but this can greatly increase the conduction of heat through the ITPS. On the other hand most insulation materials such as ceramics and foams have low strength. Material selection is important as well as dimensioning the structure properly, transforming this design into a multidisciplinary optimization problem. Even assuming that the material properties and their exact behavior are known accurately, this problem is computationally intensive involving large number of finite element analyses. If composites are used for ITPS, it further increases the number of uncertainties in the problem. When these uncertainties are included in the design, the computational expense becomes prohibitively large. This could be solved using traditional deterministic 15

16 methods, which are computationally feasible but less accurate. This problem could be definitively addressed through probabilistic optimization, which is accurate but computationally intractable. The overall objective of this dissertation is to employ efficient and cost effective methods to optimize the ITPS panel. Various techniques have been applied along the different phases of this dissertation to increase the computational efficiency and reduce the cost and time involved. To reduce the computational cost of structural analysis of ITPS, finite element based homogenization is investigated. Homogenization is a process of approximating the behavior of heterogeneous structures as homogeneous by determining their equivalent stiffness properties. Past research efforts have shown that homogenization of the composite structure as an equivalent two-dimensional orthotropic plate would reduce the computational cost associated with structural analysis tremendously [1, 2, 3, 4, 5]. Sharma et al. [6, 7] applied the homogenization to an ITPS panel with corrugated core. Owing to negligible foam density, the panel was modeled using shell elements. The ITPS panel considered in this dissertation has denser insulation foam requiring 3-D brick elements. This research thus aims to extend the homogenization technique to 3-D sandwich panels. In homogenization, the equivalent stiffness properties of a composite sandwich panel are the in-plane stiffness [A], coupling stiffness [B], bending stiffness [D] which are combined and represented as the [ABD] stiffness matrix and transverse shear stiffness (A 44 and A 55 ). The equivalent stiffness properties were determined by taking advantage of the repeating structure of the panel. The repetitive block of the panel is referred as unit cell or representative volume element (RVE). The [ABD] stiffness matrix was determined by subjecting the unit cell to unit strains and curvatures. The transverse shear stiffness was determined by analyzing a one-dimensional plate of the ITPS model under end loads. However, it was found that homogenization of the composite sandwich panel was accurate only when the panel dimensions are much 16

17 larger than the characteristic dimensions of the unit cell. Hence one unit cell of ITPS panel is considered and analyzed during the optimization process. The second objective of this research is to perform design optimization through deterministic and reliability based approaches. One of the main purposes of optimization is to minimize the weight of the ITPS while satisfying several conflicting design constraints. Design optimization through a traditional, safety factor approach without considering the uncertainties involved would provide less accurate and conservative designs. The composite panel under consideration contributes to the major part of the weight on the exterior of the space vehicle. Thus it is imperative to reduce the structural weight of ITPS without resulting in conservative and unsafe designs. Hence, it is desired to optimize the design of the ITPS using reliability based approach by including all the uncertainties in the design. However it is desired to perform an initial deterministic optimization as it would help in narrowing to the appropriate design space to perform reliability based optimization. Like most engineering structures, the ITPS could undergo multiple failure modes: temperature, stresses and buckling loads. For deterministic optimization, the design variables are optimized to minimize the structural mass. The optimization process involves evaluation of the aforementioned failure inducers at large number of design points which increases the computational cost. Construction of surrogate models (interchangeably referred as response surface approximation) of the constraints at limited design points serve as an efficient alternative to high fidelity finite element analyses. Reliability based optimization has an additional reliability constraint that is a function of design variables and random variables. The design variables are dimensions of the ITPS panel and the random variables considered are material properties, geometry, thermal and mechanical loads. Material properties are complex to model as they depend on temperature. Geometry is another source of uncertainty affecting both the structural behavior and thermal loading. Hence it is important to 17

18 analyze the failure inducers such stresses, temperature and buckling due to the above mentioned uncertainties. With large number of uncertainties involved, even construction of quality surrogate models requires finite element analyzes at extremely large number of random inputs. Failure constraints that separate the feasible designs from the infeasible ones have to represented accurately in order to determine an accurate optimum. Commonly the constraints are evaluated using global surrogate models using space filling design of experiments. This would result in wasting a large number of high fidelity finite element analyses. Earlier research efforts have introduced sampling methods to improve accuracy of the target regions with less number of designs [8, 9, 10, 11, 12, 13, 14].The sampling method developed by Bichon et al. [14], Efficient Global Reliability Analyses (EGRA) sequentially samples design points in the vicinity of the target region using a gaussian process response surface constructed initially with fewer design points. This could drastically reduce the number of computationally expensive finite element simulations while resulting in accurate optimum design. The ITPS optimization problem involving high dimensional design space seeks polynomial response surface. This research work extends the current EGRA methodology to adapt to the current ITPS problem by incorporating polynomial surrogates in the algorithm. The goal of reliability analysis is to determine the probability that a system will fail in service, given that its behavior is dependent on random inputs. To calculate the system reliability (probability of failure), Monte Carlo simulation (MCS), a commonly used method for multiple failure modes is employed. To efficiently handle numerous uncertainties and multiple failure modes, Separable Monte Carlo (SMC) developed by Smarslok et al. [15] was adopted. When the uncertain random variables are statistically independent, the concept of SMC is to group the uncertainties involved in the problem and sample them appropriately to have an accurate failure prediction for a given computational budget. Previous work with SMC only explored simple limit states 18

19 expressed as a difference between a random response and a random capacity [16]. This research work looks at a more general limit state function that combines sets of random response and capacity components [17]. Further the error in the probability of failure estimate for the simple limit state was derived in terms of the number of samples of the response and capacity. This is not applicable for the generalized limit state. This research aims at estimating the error in the probability of failure for separable case using the bootstrapping method [18], a resampling technique, which involves taking the samples of response (expensive) and resampling them with replacement [19] Motivation and Objectives below. 1.1 Outline of Dissertation To summarize the above discussion, the main objectives of this research are given Optimization of ITPS is computationally expensive, because of Structural analysis of high fidelity 3D finite element model Probabilistic analysis including all uncertainties - aleatory (material properties, geometry and loading conditions) and epistemic uncertainties(error in modeling and simulation). Multiple failure modes of ITPS demands Monte Carlo simulations for reliability calculations. To successfully optimize the ITPS structure through deterministic and reliability based optimization, this dissertation proposes To develop a reduced but efficient model of ITPS to lower the computation cost involved in the finite element analyses. To investigate low fidelity surrogate models (Response Surface Approximation of the constraints) to replace high fidelity finite element analyses. To adapt Efficient Global Reliability Analysis that facilitates sequentially sampling around the constraints requiring fewer number of samples to improve target region and predict accurate optimum. 19

20 To utilize separable Monte Carlo method that allows separable sampling of the expensive and inexpensive variables which will reduce computational cost and also provide improved accuracy. This dissertation essentially demonstrates the methodology for deterministic and reliability based optimization of complex structures with large number of uncertainties using surrogate models, EGRA, separable Monte Carlo technique and bootstrapping method Thesis Organization Chapter 2 reviews the thermal protection system used in space vehicles such as Challenger, Discover and Columbia. This chapter presents a review on the finite element based homogenization technique. Further, presents a review on structural optimization and the different aspects of deterministic and reliability based optimization. Chapter 3 presents detailed description on the geometry, components, loads and boundary conditions of the multi-functional Integrated Thermal Protection System followed by a discussion on finite element analyses of ITPS involved in the design and the optimization process. Chapter 4 present the finite element based homogenization method and results obtained from the method. Chapter 5 presents a detailed discussion on Monte Carlo simulation techniques to estimate probability of failure which includes crude Monte Carlo and separable Monte Carlo method. bootstrapping method to estimate error in the probability of failure applying it to a pressure vessel problem. Chapter 6 illustrates the working of Efficient Global Reliability Analysis (EGRA) using a two variable problem. EGRA applied to the optimization of the ITPS panel is discussed in Chapter 7. This chapter presents the deterministic and reliability based optimization of the ITPS panel and other important components (uncertainty modeling and propagation, sensitivity analyses) associated with the optimization process. The dissertation is concluded with summary of results and possible future work in Chapter 8. 20

21 CHAPTER 2 BACKGROUND STUDY AND LITERATURE REVIEW Thermal protection systems have been an area of continuous research and development in space vehicle design. With atmospheric heating during reentry being the major cause of concern for a space vehicle s safe operation, several design and analysis techniques have been put forth by researches. This chapter reviews the different types of Thermal Protection system used so far in space vehicles. Further it also introduces and reviews the various techniques that are aimed at reducing the computational cost associated with the deterministic and reliability based optimization of the structurally integrated Thermal Protection System. 2.1 Orbiter Thermal Protection System-30 years Legacy When a space shuttle orbiter re-enters the Earth s atmosphere, it is traveling in excess of 17,000 mph ( 7600 m/s). While slowing down to landing speed, friction with the atmosphere produces external surface temperatures as high as 1900 K. Special thermal shields on the exterior surface protect the vehicle and its occupants during launch and reentry [20, 21, 22]. Earlier manned spacecraft, such as Mercury, Gemini and Apollo, were protected during re-entry by a heat shield constructed of phenolic epoxy resins in a nickel-alloy honeycomb matrix. The heat shield was capable of withstanding very high heating rates. During the reentry, the heat shield would ablate, or controllably burn with the char layer protecting the layers below. Despite the advantages, ablative heat shields had some major drawbacks. They were bonded directly to the vehicle, they were heavy, and they were not reusable. With a design life of 100 missions, space shuttle orbiter required a light weight reusable thermal protection system (TPS) not only to protect the orbiter from the searing heat of reentry, but also to protect the airframe and major systems[23, 24]. The vehicle s configuration and entry trajectory defines the temperature distribution on the vehicle. Over the past 30 years, the Space Shuttle s Thermal Protection system 21

22 has been built with materials with a high temperature capability and underlying thermal insulation to inhibit the conduction of heat to the interior of the vehicle. During the entire mission, different locations on the orbiter get heated to different temperatures. The leading edges of wings and the nose cap are the highest temperature regions. Due to the wide variation of these temperatures the TPS selected for space shuttle was composed of many different materials. Each material s temperature capability, durability and weight determined the extent of its application on the vehicle. There are basically two categories of TPS being used, the tile TPS and non-tile TPS. A detailed description of each category would provide a better understanding on the TPS selection for different regions on the vehicle [25, 26]. Figure 2-1. Thermal Protection System (TPS) used in space shuttles [26] 22

23 2.1.1 Tile TPS The different types of tile TPS are High-temperature reusable surface insulation (HRSI) tiles were developed to provide protection against temperatures up to 1533 K. They were used in areas on the upper forward fuselage, vertical stabilizer leading edge, and upper body flap surface. The tile is composed of high purity silica fibers. A black coating, reaction cured glass (RCG) was applied to all but one side of the tile to protect the porous silica and to increase the heat sink properties. HRSI was primarily designed to withstand transition from areas of extremely low temperature, about -270 C to the high temperatures of re-entry typically around 1600 C thus maximizing heat rejection during the hot phase of reentry. Fibrous refractory composite insulation tiles (FRCI) were similar form of the HRSI tiles. The FRCI tiles had higher strength derived by adding alumina-borosilicate fiber, called Nextel, to the pure silica tile slurry. Though developed for the same purpose, FRCI and HRSI had different physical properties because of 20% Nextel in it. FRCI tiles were lighter than the basic HRSI tiles. Furthermore, the FRCI tiles also had tensile strength that was at least three times greater than that of the HRSI tiles and could be used at a temperature almost 100 C of higher than that of HRSI tiles. In a nutshell, they provided a better strength, durability, cracking resistance, and weight reduction. Toughened uni-piece fibrous insulation (TUFI) is an improved low density rigid ceramic composite, with very high impact resistance ( times more than RCG coating). TUFI was used in regions where temperatures reach as high as 1260 degrees Celsius. TUFI tiles were built as high temperature black versions for use in the orbiter s underside providing sufficient heat insulation for the orbiter s underside. And lower temperature white versions for use on the upper body conducting more heat which limits their use to the orbiter s upper body flap and main engine area. Low-temperature reusable surface insulation (LRSI) were white in color possessing high thermal reflectivity. These tiles were used to protect areas where reentry temperatures are below 649 C (1200 F). They are generally installed on the upper surface of the vehicle, maximizing solar gain when the orbiter is on the illuminated part of the orbit. They were also used to protect the upper wing near the leading edge and also areas of the forward, mid, and aft fuselage, vertical tail Non - Tile TPS The different types of non-tile TPS are Reinforced Carbon-Carbon (RCC) laminated composite material were primarily used for covering the wing leading edges and nose cap where the temperatures 23

24 reach a maximum of 1510 C during reentry. This composite covering had very high fatigue resistance which is essential during ascent and reentry. Generally all TPS components (tiles and blankets) were mounted onto structural materials that support them, mainly the aluminum frame and skin of the orbiter. RCC is the only TPS material that also served as a support for parts of the orbiters aerodynamic shape, wing leading edges and the nose cap. Flexible Insulation Blankets (FIB) were white low-density fibrous silica material. These blankets were developed as replacement for LRSI tiles. They require very less maintenance than LRSI tiles yet they possessed the same thermal properties. Felt Reusable Surface Insulation (FRSI) are generally white, flexible fabric offering protection up to 371 C (700 F). FRSI covered the Orbiter s wing upper surface, the upper payload bay doors, and aft fuselage. Gap fillers were placed at doors and moving surfaces to minimize the heat created open gaps in the heat protection system. These materials were used around the leading edge of the forward fuselage nose caps, windshields, side hatch, wing, vertical stabilizer, the rudder, body flap, and heat shield of the shuttle s main engines. The filler materials are made of either white AB312 fibers or black AB312 cloth covers (which contain alumina fibers). For further in depth discussion on thermal protection concepts used in space vehicles, the reader is referred to Blosser [27] and Bapanapalli [28]. 2.2 Finite Element based Homogenization Micromechanical analyses have been traditionally used to estimate the effective stiffness properties of composite materials. Some of these methods are also suitable to obtain homogenized properties for plate like structures with periodicity. There are several approaches to homogenization: mechanics of materials approach, elasticity approach, energy methods and finite element analysis. All methods assume that there is a representative volume element (RVE) or unit cell that repeats itself to form the structure. The unit cell is subjected to six linearly independent deformations to determine the equivalent stiffness properties of the panel. The deformations are applied as periodic displacement boundary conditions (PBC) which includes three mid-plane strains and three curvatures. 24

25 Biancolini [1] used an energy equivalence approach to homogenize a corrugated core panel to an equivalent anisotropic lamina. The homogenization procedure was applied using static condensation in which the internal nodes of the micro geometry are removed and external nodes at the boundary of the model represent the edges of the equivalent lamina. The effective stiffness properties obtained using this method was validated with a series of numerical tests. Yu et al. [29, 30] variational asymptotic method for unit cell homogenization (VAMUCH) to predict effective properties of periodically heterogeneous materials. Davalos et al. [2] evaluated the equivalent properties of fiber reinforced honeycomb sandwich panels (Figure 2-2). The panel consisted sinusoidal core in the plane and extended vertically between the laminates. The homogenization technique was a combination of energy method and mechanics of material approach to predict the equivalent properties of the panel. For a similar sinusoidal core panel, Buannic et al. [3] used an asymptotic expansion method for estimating the equivalent stiffness properties. Wallach and Gibson [4] used the unit cell approach to calculate the stiffness properties, compressive strength and shear strength of sandwich structures having pyramidal truss cores; employing two methods, a truss-analysis program based on matrix methods and a commercial finite element analysis (FEA) program using ABAQUS. The finite element approach confirmed results from the truss analyses and allowed to extend the analysis to include nonlinear effects such as material yield and large deformation. Past efforts on the homogenization of the ITPS panel (Figure 2-3) include an analytical approach and finite element based approach. Martinez et al. [5, 31] followed strain energy approach and shear deformable plate theory to develop an analytical model for the homogenization of a corrugated sandwich panel (Figure 2-3) of the ITPS. Though analytical models provide reasonably good estimate of stiffness properties, they involve several assumptions compromising the accuracy of the structure. Moreover most of the analytical approaches require a finite element based validation. Sharma et al. 25

26 Figure 2-2. Sinusoidal core sandwiched between laminates [1] [32, 6] employed a finite element based homogenization technique for homogenizing corrugated core sandwich panel as an orthotropic plate and showed that the results agreed well with the three dimensional model. Hence a finite element method based homogenization procedure is adopted here to obtain the equivalent plate properties. In finite element based homogenization periodic boundary conditions are imposed on the representative unit cell (RVE) that corresponds to a given state of mid-plane strains and curvatures of the equivalent plate to obtain the effective stiffness properties. The finite element based homogenization of the ITPS panel (Figure 3-1) considered in this study will be discussed in Chapter 4. Figure 2-3. Integrated Thermal Protection System with corrugated core[28, 31] 26

27 2.3 Deterministic and Reliability based optimization An efficient structural design plays an important role in various engineering disciplines (for eg. in automobile design, aerospace engineering and in marine applications). A key area of continuous research is in finding an optimum, efficient structure - making a structure light weight yet having adequate load carrying properties, is a popular example. Recent development in optimization theory and computational tools have facilitated ways to find optimal structures [33]. Further, design and optimization of composite structures have gained special attention [34, 35, 36]. The methodologies discussed in this research are deterministic optimization and reliability based optimization. Deterministic optimization involves minimizing the weight while applying safety factors on failure constraints. Reliability based optimization includes all the randomness in the design process and has an additional constraint referred as reliability or probability of failure constraint. Reliability based optimization has gained increasing popularity mainly due to the use of composite materials and the typical design drivers being minimum structural mass and high reliability. However it proves beneficial to perform deterministic optimization initially as it facilitates in choosing appropriate design space for reliability based optimization. The optimization process involves evaluation of the failure constraints using finite element analyses that are computationally expensive. This research proposes to perform deterministic and reliability based optimization of the integrated thermal protection system composite panel and aims at reducing the computational cost at different phases of optimization, using techniques such as response surface methodology, Efficient Global Reliability Analysis (EGRA), separable Monte Carlo (SMC) method and bootstrapping method. The following sections presents some background and review on deterministic optimization, use of surrogate models for optimization, types of uncertainties included to estimate reliability, methods to evaluate probability of failure, methods to estimate accuracy of probability of failure. Further it also discusses Efficient Global Reliability 27

28 Analysis (EGRA) for accurate estimation of the optimization constraints with limited computational resources Deterministic Optimization Deterministic optimization involves optimizing the design variables to minimize the objective function while satisfying equality or inequality constraints that follow a linear or non-linear formulation. The general formulation of the problem is Minimize such that w = f(d) g(d) z h(d) = m (2 1) d LB < d < d UB where the objective function f is a function of the design variables d, subject to in equality constraints g and equality constraints h, both function of d and LB and UB are lower bound and upper bound on the design variables. z and m are the allowable thresholds of the constraints. In structural applications, the design variables are geometry of the structure, objective function is the structural mass and constraints are safety factor based failure constraints such as deflection, stresses, buckling and temperature. Since it is a safety factor based approach, deterministic optimization often results in conservative and unsafe optimal designs. However, when the bounds of the design variables are not certain, an initial deterministic optimization helps in narrowing the design space with more feasible designs to perform reliability based optimization. For the deterministic optimization of the ITPS panel, the design variables are the dimensions, t W, t B, t T, and h. The complex design and conflicting requirements of the ITPS panel leads to various failure modes a) the maximum bottom face sheet temperature, b)maximum stresses in the ITPS components and c) buckling load. It is optimized to minimize the structural mass while satisfying thermal, structural and buckling constraints. Safety factors are applied on each failure constraint. A factor of 28

29 77K on the temperature constraint and 1.5 on both the stress and buckling constraints. Since the deterministic optimum will lie on either or all of the constraints, it separates the feasible design space from the infeasible one. This helps in narrowing the design space to the feasible ones for performing probabilistic optimization and thus avoiding expensive FE analysis on the infeasible space Structural Reliability Reliability based optimization has grown rather quickly during the last few decades and structural reliability methods have developed rapidly and have been widely applied in the practical design of structures [37]. The aim of reliability analysis is the quantification and treatment of uncertainties and then the evaluation of a measure of safety or reliability to be used in design [38]. The terms reliability and probability of failure are complementary, in that the more reliable the design, the lower the probability of failure. In structural applications, reliability based optimization involves minimizing weight of the system while satisfying a reliability/probability of failure constraint. The general formulation of the problem is Minimize such that such that w = f(d) p f (g(d, X) < z) < p f,cr or β(g(d, X) < z) > β Cr (2 2) where the objective function f is a function of only the deterministic design variables d, but the response function in the probability of failure constraint g is a function of d and X, a vector of random variables defined by known probability distributions [39, 40]. p f,cr is the target probability of failure. Since probability measures are highly non-linear, this constraint is often replaced by reliability constraint where the reliability index β is calculated as 29

30 β = Φ 1 (p f ) (2 3) where Φ is the cumulation distribution function of standard normal distribution. β Cr is the target reliability index. Reliability analyses can be performed using Moment-based methods such as the first-order-reliability-method (FORM) or second-order-reliability-method (SORM) or sampling methods such as Monte Carlo simulations (MCS). Researchers have used these methods to optimize composite structures as well. Leheta and Mansour [41] carried out limit state analysis, first order reliability analysis and reliability- based structural optimization of ship stiffened panels. Qu and Haftka [35] demonstrated reliability based optimization of composites at cryogenic temperatures using response surface approximations and Monte Carlo Simulations. Scuiva and Lomario [42] performed a comparison between Monte Carlo simulations and FORMs in calculating the reliability of a composite structure. A significant amount of research has dealt with optimization and reliability analyses of the corrugated core ITPS panel (Figure 2-3). Kumar et al. [43] and Villanueva et al. [44] studied the difference in risk allocation between structural and thermal failures by deterministic and probabilistic optimization. Sharma et al. [45] performed a multi-fidelity analysis of the ITPS panel using Correction Response Surface. Further Villanueva et al. [46] studied the effects of single future test in the deterministic and probabilistic design of the ITPS panel. They considered only the thermal failure mode of the ITPS panel. Matsmura et al. [47] considered multiple failure modes of the ITPS panel to study the effect of future tests in the reliability estimation of the ITPS panel. This research considers a different concept of ITPS and aims at deterministically and probabilistically optimizing the ITPS panel using cost effective adaptive sampling techniques. Moment based methods can be cost-efficient as they involve approximation of response function to find the most probable point (MPP), however, they are inaccurate. 30

31 Sampling methods are accurate if adequate samples are used but computationally intensive. In engineering applications reliability estimation may involve individual failure mode such as stress or system level failure with multiple failure modes which could be, say, stress and deflection. Moment-based methods could be used for system with single failure modes. However when the problem involves higher dimensions with multiple failure modes that are expensive to evaluate, the response functions cannot be approximated as analytical derivatives [42]. Such situations are addressed by numerical approximation of responses using surrogate models and sampling methods such as Monte Carlo simulations to estimate reliability. As mentioned earlier, the ITPS panel considered has multiple failure modes and involves calculation of system level reliability. System level failure is defined either as parallel failure or series failure. When system is assumed to have failed if all the modes fail it is referred as a parallel system failure, when either one of the modes fail, it is referred series system failure. In the ITPS problem, if either of the modes, temperature, stress or buckling modes fail, the panel is considered to have failed. 2.4 Use of Surrogate Models In order to reduce the computation cost associated with finite element simulations, quality surrogate models of the various constraints are constructed. They are alternately referred as Response surface approximations (RSA) or metamodels. Surrogate models usually employ low order polynomials to the structural response with a limited number of input design points [48, 49]. The initial design points are obtained using design of experiments such as full factorial design, Central composite design, Latin hypercube, A-optimal or D-optimal designs. In the case of expensive response evaluations, surrogate models have been used for evaluating constraints in the deterministic and probabilistic optimization process. Viana et al. [50, 51] have demonstrated the use of surrogate models and surrogate based design optimization. Further they also developed a Surrogate Tool box 31

32 in MATLAB that makes the optimization process easier [52]. This research mainly uses the Surrogate Toolbox for fitting response surface models and f mincon() a MATLAB optimizer for the optimization process. Moreover, the probability calculated from Monte Carlo simulation often introduces random errors due to limited sample size resulting in unsafe optimum [35, 53]. The use of response surface approximation reduces such errors. Though surrogate models are beneficial in the case of computationally expensive problems such as those demanding complex finite element analyses (deterministic optimization), with large number of uncertainties and multiple constraints, the probability of failure calculations even with response surface models are difficult to handle. Traditionally the response surface models constructed were a global approximation of the response with design of experiments that are independent of the response function. Since failure is defined using a constraints/limit states that separate the feasible and infeasible designs, it is important to construct accurate contours of the constraints while it is acceptable to have large errors at other regions in the design space. Accurate approximation of constraints would lead to accurate prediction of optimum Application of Surrogates to Improve Constraint Boundaries In deterministic and reliability based optimization, it is important that the constraint boundaries that separate the feasible designs from the infeasible ones are estimated accurately with minimal error. Global surrogate models can be used but would require large number of design points and would not ensure accurate approximation of constraint boundaries. To build surrogate models that approximates constraint boundary accurately, researchers have introduced various methods. Kuczera and Mourelatos [8] used a combination of global and local surrogate models to first detect the critical regions and then obtain a locally accurate approximation. Arenbeck et al. [9] used support vector machine and adaptive sampling to approximate failure regions. Tu and Barton [10] used a modified D-optimal strategy for boundary-focused 32

33 polynomial regression. Vazquez and Bect [12] proposed an iterative strategy for accurate computation of probability of failure based on kriging. Shan and Wang [54] developed a rough set based approach to identify sub-regions of the design space that are expected to have performance values equal to a target value. For constrained optimization and reliability estimation, Picheny et al. [13] developed targeted Integrated Mean Square criterion (IMSE) to construct design of experiments such that the metamodel accurately approximate the vicinity of a boundary in design space which is either defined by a target value or gaussian distribution. 2.5 Adaptive Sampling When optimization problems involve non-linear and multimodal objective functions, they require large number of functions evaluations to find the global optimum. And when the evaluations are limited by expensive computations, finding an accurate global optimum becomes difficult. Surrogate models could be employed but their accuracy is also compromised when complex functions are constructed with fewer design points. Jones et al. [55] developed Efficient Global optimization (EGO), an unconstrained optimizer that focuses on adding points to the design space to accurately model complex functions and hence find the global optimum accurately. In this method, an initial Gaussian process model [56, 57] is built as a global surrogate for the response function. EGO then adaptively selects additional samples to be added to the design space to form a new Gaussian process model in subsequent iterations. EGO uses a specific formulation known as the Expected Improvement that identifies the new design points based on how much they are expected to improve the current best solution to the optimization problem. When this expected improvement is acceptably small, the globally optimal solution has been found. Efficient Global Optimization was further extended to improve target regions in the design space. Ranjan et al. [11, 58] developed a sequential design methodology based on (EGO) that explores the design along the contour of interest. Bichon et al. [59] 33

34 also developed a similar method based on EGO referred as Efficient Global Reliability Analyses (EGRA). They illustrated the application of EGRA by improving limit states for accurate reliability estimation. This research proposes to extend the application of EGRA to improve constraint boundaries in deterministic optimization and reliability based optimization of the ITPS panel. The concept of sampling designs iteratively, reduces the computational cost of expensive FE simulations and at the same time increases the accuracy of the failure boundary Efficient Global Reliability Analysis The method has two main features, an initial gaussian process model and the expected feasibility function to identify the additional samples iteratively at the vicinity of the limit state. The initial surrogate model Ĝ is constructed using limited number of designs using a known sampling method, in this case, Latin hyper cube sampling. The expected feasibility function is optimized to find the next design that would improve the target boundary z. The function is given as EF (Ĝ(X)) = [ ( ) ( ) ( )] z (µ µg z µ G z + µ G G z) 2Φ Φ + Φ σ G σ G σ ( ) ( ) ( G )] z µg z σ G [2ϕ µ G z + µ G ϕ + ϕ σ G σ G σ [ ( ) ( G )] z + µ G z + µ G +ϵ Φ Φ σ G σ G (2 4) The response predicted by the gaussian model follows the distribution N(µ G, σ G ), ϵ = ασ G is the error band around the limit state which is a function of the standard deviation σ G and α, a factor applied on the standard deviation. z + = z + ϵ and z = z - ϵ. ϕ and Φ are the standard normal pdf (probability density function) and cdf (cumulative distribution function) respectively. When µ G is close to the target contour, the first term dominates the expression. These points are close to the ϵ band. If µ G is far away from target, the second term tends to dominate. This term facilitates sampling in regions of the input space where the estimated response is outside the ϵ band, but the uncertainty 34

35 of prediction is high. The third term is related to the variability of the predicted response in the neighborhood of ϵ [11, 58]. It supports sampling in regions near the estimated contour but where the prediction variance is quite high. Currently either gaussian process or kriging are the default surrogate models for implementing EGRA. Since the ITPS involves a large number of random variables, the use of polynomial response surface is favored over gaussian process or kriging. Hence to adapt to the optimization of the ITPS panel this research proposes to implement polynomial response surface in EGRA methodology. 2.6 Uncertainty Modeling Modeling and analysis of input uncertainties aid in understanding the propagation of uncertainty in the outputs. Uncertainties in an engineering system can be classified as epistemic uncertainty and aleatory uncertainty also called variability. Epistemic uncertainty generally represents a lack of knowledge of a quantity or process of a system or environment.this uncertainty is also referred as reducible uncertainty as it can be reduced (or increased) from increased understanding of the uncertain variable or from more relevant experimental data. Aleatory uncertainty is generally characterized by inherent randomness in the physical system or environment which cannot be reduced by further data [60]. In structural applications, aleatory variability can be introduced by manufacturing imperfections such as variability in material properties and geometric dimensions, variability in loading and epistemic uncertainties by errors in modeling and simulation. Uncertain variables are included in the design process by their respective probabilistic distributions. In addition to the variability in the input parameters, error in modeling and simulation (epistemic uncertainty) is also introduced in the output parameters to estimate the probability of failure. The probability distribution of the random variables could follow uniform, normal, lognormal or Weibull distribution [61, 15]. However, this research mainly assumes uniform and normal distribution for the input random variables. 35

36 Estimates such as variance, standard deviation or coefficient of variation of the response samples would provide the uncertainty in the response due to input uncertainties. The importance of uncertainty characterization and uncertainty propagation in composites has been illustrated by various researchers[62, 63, 64, 65, 66]. António and Hoffbauer [62] studied the effects of deviations in mechanical properties, ply angles, ply thickness and applied loads for a laminated shell composite. They demonstrated that uncertainty analysis is very useful in designing laminated composite structures minimizing the unavoidable effects of input parameter uncertainties on structural reliability. Oh and Librescu [63] addressed the problem of alleviating the effects of uncertainties for free vibration of composite cantilevers under uncertainties such as layer thickness, elastic constants and ply angle. Noor et al. [64] studied the variability of non-linear response of stiffened composite panels due to variations in geometric and material parameters using hierarchical sensitivity analysis and fuzzy set analysis approach. For the ITPS optimization problem, the random variables are dimensions, mechanical loads, thermal loads, thermo-mechanical properties such as Young s modulus, shear modulus, poisson s ratio, coefficient of thermal expansion, thermal conductivity and specific heat. In addition to this, uncertainty is considered in the allowable limits such as material strength, temperature. The error in finite element modeling and simulations (transient heat transfer, static stress and buckling analyses) is also included. 2.7 Estimation of Probability of Failure The probability of failure is the probability that the random variables X = {x 1, x 2,...x i } are in the failure region that is defined by limit state function G(X) < 0. If the joint pdf of X is f x (X),the probability of failure is evaluated with the integral p f = P {G(X) > 0} = f x (X)dx (2 5) G(X)<0 36

37 when The reliability is computed by R = 1 p f = P {G(X) > 0} = 1 G(X)<0 f x (X)dx (2 6) The direct evaluation of the probability integration state above is extremely difficult a large number of random variables are involved. In such cases the probability integration becomes multidimensional which is typical of engineering applications. the integrand f x (X)is the joint pdf of X and is generally a nonlinear multidimensional function. the limit state boundary G(X)=0 is also multidimensional, nonlinear function. In structural applications, G(X) is often a black-box model (or finite element simulation), and the evaluation of G(X) is computationally expensive. Hence moment based methods and sampling methods could be applied to evaluate the probability integration Moment based Methods Moment based methods such as First-Order Reliability Method (FORM) and Second-Order Reliability Method (SORM) ease the computational difficulties by simplifying the integrand f x (X) and approximating the limit state function G(X) to estimate probability of failure (Equation 2 6). The method simplifies the joint distribution function by transforming the original random variables from X-space to standard normal U-space. The limit state function is approximated using first-order Taylor series expansion. In the standard normal space, the point on the limit state function where G(U) = 0 at the minimum distance from the origin is the most probable point (MPP) of failure. The reliability is measured as the distance from the origin to the MPP. This measure is referred as reliability index (β). The MPP is determined as Minimize β = U T U such that G(U) = 0 (2 7) 37

38 where U is the vector of variables in standard normal space. FORM is fairly accurate when the limit state function can be approximated as a linear function. Second order methods can be used when the limit state function has a higher order of curvature. This method approximates the limit state as a quadratic, and provides a more accurate approximation in such cases Sampling Methods Monte Carlo simulation (MCS) is one of the powerful and easy to implement methods to propagate the uncertainty in input random variables to the uncertainty in failure [67]. Reliability-based design of a structural system is often addressed using Monte Carlo simulations [68, 39, 69], especially when the system under consideration fails due to multiple failure modes. Probability of failure p f is determined through a limit state function G that separates the feasible designs from the infeasible ones. The limit state is generally a function of random variables, response R and capacity C. When the response exceeds the capacity, it is considered failure, example, maximum stress failure theory where the response will be calculated stress and capacity would be strength of the material. The capacity and response are assumed to be functions of statistically independent random variables X 1 and X 2, respectively. Equation 2 8 shows the separable case where failure occurs when a single component of response exceeds a single component of capacity. G(X 1, X 2 ) = R(X 1 ) C(X 2 ) (2 8) Failure occurs when G 0 and the system is safe when G < 0. In the more general case, the capacity and the response in the limit state cannot be explicitly separated, and the limit state function may be represented as G(X 1, X 2 ) = G(R(X 1 ), C(X 2 )) (2 9) 38

39 Where R and C may be scalar or vector quantities. The limit state could be a higher order polynomial function in terms of response and capacity such as Tsai-Hill or Tsai-Wu criterion generally employed to predict failure in composite structures. Statistical estimates such as variance, standard deviation and coefficient of variation of probability of failure would provide accuracy in the estimate. The traditional, crude Monte Carlo technique (CMC) is simple, but it lacks accuracy when sampling is limited due to computationally expensive structural analysis, such as from finite element analysis (FEA). There are various techniques to improve the accuracy or efficiency of CMC, including tail modeling, conditional expectation, importance sampling and the use of surrogates [70, 71]. However, another technique referred as separable Monte Carlo was developed by Smarslok et al. [72, 16], which is applicable in combination with the above mentioned methods to further improve accuracy or efficiency. When the response and capacity random variables are statistically independent, accuracy can be improved by the separable Monte Carlo method (SMC). This facilitates improved accuracy of the calculation of the probability of failure for the same computational budget. The error in probability of failure estimate for crude Monte Carlo can be obtained from binomial law. For SMC involving simple limit state (Equation 2 8), the variance estimator was derived using conditional calculus and validated using simulation estimates [15, 16]. The simulation estimates of variance were shown to be of comparable accuracy to those obtained for CMC. However this is not applicable in the case of complex non-separable limit state (Equation 2 9). For SMC with non- separable limit states, bootstrapping technique is proposed [18]. The error in the standard deviation estimate of the bootstrapped probability of failure provides a measure of the accuracy of bootstrapping. Further it was also demonstrated that the variability estimate of capacity and response can help in choosing the sample size needed for given accuracy[17, 19]. Chapter 5on estimation of probability of failure provides a detailed description of the 39

40 Crude Monte Carlo, Separable Monte Carlo method, bootstrapping technique and regrouping of random variables to improve accuracy of p f by applying these concepts to a composite pressure vessel problem. 40

41 CHAPTER 3 DESIGN AND ANALYSIS OF INTEGRATED THERMAL PROTECTION SYSTEM PANEL 3.1 Design of ITPS Composite Panel The primary objective of this research, as stated earlier, is to determine the optimal design of an ITPS composite panel. This chapter discusses the geometry and material selection involved in a typical ITPS construction. Further, the finite element analyses of the ITPS model is also discussed. The integrated Thermal Protection Sytem panel considered here consists of stacked rigid Alumina Enhanced Thermal Barrier foam (AETB) insulation bars that are spirally wrapped with a Silicon Carbide/Silicon Carbide (SiC/SiC) laminate. The bars are stacked orthogonally in two layers in a 0/90 configuration. The bars are then supported by a top face sheet and a bottom face sheet made of SiC/SiC and Graphite/Epoxy laminate, respectively forming a sandwich structure (Figure 3-1). The various components, stack orientation and material properties are given in Tables 3-2 and 3-3 [73, 74, 75]. The concept not only has material asymmetry about the mid-plane but also geometric asymmetry as the rigid insulation bars are stacked orthogonally. Further, the design is complex because of the laminate wrapped around the AETB insulation bar. It is important to note that in such a design, transverse shear of the insulation and the webs (wraps) would have a pronounced effect on the structure Key Dimensions of the ITPS structure The key dimensions of the ITPS are the width w f, and height h of the insulation foam, thickness of the top face sheet t T, bottom face sheet t B and wrap t W and the number of bars n (Figure 3-2). The nominal values of the dimensions are given in Table 3-1. Table 3-1. Key Dimensions of the ITPS panel w mm h mm t T mm t B mm t W mm n

42 Figure 3-1. Schematic view of the ITPS panel Figure 3-2. Dimensions of the ITPS panel ITPS Materials The components of ITPS are made of ceramic matrix composites, polymer matrix composite and high density insulation foam. The top face sheet is made of SiC/SiC plain woven textile composite laminate which is known for its high mechanical strength at elevated temperatures, high thermal stability and low density [76]. The bottom face sheet is graphite/epoxy laminate (polymer matrix composite). The notable properties of polymer matrix composites are high tensile strength and stiffness, high fracture toughness and good corrosion resistance. Moreover they are popular due to their 42

43 low cost and simple fabrication methods. The insulation, made of alumina enhanced thermal barrier (AETB) foam demonstrate higher strength, added durability, and have a maximum operational temperature of 1700K [26]. The material properties of the components, the fiber orientation and the thickness of each component of ITPS are presented in Table 3-2 and Table 3-3 below. Table 3-2. Components of ITPS, materials and fiber orientation Component Material Fiber orientation Thickness Top face sheet SiC/SiC laminate [(0/90) 4 ] S 16 layers mm each Insulation foam AETB 8-35 mm Bottom face sheet Gr/Ep laminate [(0/90) 4 ] S 16 layers mm each Wrap SiC/SiC laminate [0/90] S 4 layers mm each Table 3-3. Material properties of the components of ITPS [75, 74, 73] Properties SiC/SiC Graphite/Epoxy AETB foam E 1 GPa E 2 = E 3 GPa v 12 = v v G 12 =G 13 GPa G 23 GPa α 1 x10 6 /K α 2 x10 6 /K Boundary Conditions In the past research on ITPS panels, the common boundary conditions applied on the ITPS panel were simply supported boundary conditions and clamped boundary conditions. Also, since the panel is symmetric in the x and y directions, only one quarter of the panel is considered and symmetry boundary conditions were applied. In this research work, simply supported boundary conditions and fixed boundary conditions are considered for the homogenization of the panel (Chapter 4). However it was found that homogenization could not be satisfactorily applied to the ITPS panel (detail discussion in Chapter 4). Hence a new set of boundary conditions which simulates a structurally integrated TPS panel was analyzed. It simulates the panel mounted on the frame of the vehicle and allowed to expand equally in the x y plane. 43

44 Figure 3-3. Boundary conditions applied on the ITPS panel Loads From take off to landing, the space vehicles are subjected to various loading conditions such as drag, aerodynamic heating (thermal loads), in-plane inertial loads (mechanical loads), pressure and foreign object impact loads. These loads could act individually at any time during the mission, or they could act together leading to multiple component failure. This research work primarily explores and analyzes the effect of thermal loads caused by temperature gradient and mismatch in thermo-mechanical properties of different material combinations, and mechanical loads (pressure) through transient heat transfer analyses, thermal stress analyses, pressure analyses and thermal and mechanical buckling analyses. 3.2 Finite Element Analyses This section presents a description of the finite element models and analyses for heat transfer, stress analysis and buckling analysis of the ITPS panels. The model considered in this research is constructed using a commercial finite element software, ABAQUS (Figure 3-3). ABAQUS. Twenty-node three-dimensional brick elements were used to model the ITPS. The 3D solid element model has 3 displacement degrees of freedom at each node. 44

45 3.2.1 Transient Heat Transfer Analysis It was observed that the reentry heating rates are more severe than the heating rates during ascent, that is, the heating rates increase more steeply and the total integrated heat load is much larger during reentry. Thus, it can be inferred that the reentry heating rates would be most influential in the ITPS design and, therefore, they were used in the heat transfer FE analysis for the design process. A typical heating rate used for design is shown in Figure 3-4. The initial temperature of the structure is assumed as 295 K (72 F) [28]. A large portion of this heat is radiated out to the ambient by the top surface. The remaining heat is conducted into the ITPS. Some part of this heat is conducted to the bottom face sheet by the insulation material and some by the webs. Figure 3-4. Heat flux profile incident on the ITPS panel Loads and boundary conditions for the heat transfer problem are schematically illustrated in Figure 3-5. The four outer faces and bottom surface of the bottom face sheet are assumed to be perfectly insulated. This is a worst case scenario where the bottom face sheet temperature would rise to a maximum as it cannot dissipate the heat. It is also assumed that there is no lateral heat flow out of the unit cell. The heat flux incident on a unit cell is completely absorbed by that unit cell only, though in an actual 45

46 ITPS panel heat would flow into the face sheets and wraps which could act as a thermal mass and there would be a lateral flow of heat in the panel from one unit cell to another. The load steps are tabulated in Table 3-4. Figure 3-5. Boundary conditions for the transient heat transfer analysis Table 3-4. Heat flux load steps in the transient heat transfer analysis Load Time Heat Flux Time Step Ambient Step Period Input Size Temperature Step ,069 W/m 2 30 sec 213 K Ramp linearly Step ,06939,748 W/m 2 25 sec 243 K Ramp linearly Step , W/m 2 30 sec 273 K Ramp linearly Step sec 295 K The first three load steps considered in the heat transfer analysis, the heat flux is ramped linearly from 0 to 34,069 to 39,748 W/m 2. The step 4 represents the time period after touchdown and the FE analysis continues for another 50 seconds in order to capture the temperature rise of the bottom face sheet. During this period, along with radiative heat transfer, convective heat transfer boundary conditions are imposed on 46

47 the top surface to simulate the heat transfer to the surroundings while the vehicle is standing on the runway. The maximum bottom face sheet temperature is recorded from this analysis. Also, along the reentry time the temperature distribution with maximum difference between the top face sheet and bottom face sheet would induce maximum stresses in the panel (Figure 3-6). Hence this distribution along the height of the panel is recorded which acts as the thermal load for the stress and buckling analyses. Figure 3-6. Temperature variation on TFS and BFS with respect to reentry time Stress Analysis Stresses developed in the ITPS due to thermal and pressure loads are analyzed individually by performing a static stress analysis. In previous studies [7, 28, 31] the ITPS panel was assumed to be simply supported along its edges. It was found that the deformations due to thermal stresses were not realistic as there was no compatibility of displacements between adjacent panels. In order for an ITPS to be effective they have to be structurally integrated (S-ITPS). 47

48 In this concept the attachments will be designed such that the thermal forces will be transferred from one panel to next adjoining panel. Thus one can impose periodic BCs between panels as long as the surface heating is uniform. The problem can be further simplified by analyzing one unit cell with periodic BCs. This is is possible because we assume each panel is composed of repeating unit cells. The boundary conditions of a structurally integrated TPS panel is shown in Figure 3-3. The nodes on one of the faces on the x z plane are allowed have equal displacements in the y-direction and the adjacent face on the y z plane is allowed to have equal displacements in the x-direction. The bottom edges of these two adjacent faces were constrained from moving in the vertical direction simulating that the panel edges were mounted on the frame of the vehicle. The other two adjacent edges are assigned symmetry BCs. The temperature distributions are applied along the height of the panel (z-coordinate) on each node of the model. This implies that the top and bottom face sheet temperatures are uniform throughout the length and width of the panel. Although the temperature varies slightly in the x and y-directions, this variation is very small and can be neglected. The stresses developed in the various components ( face sheets, foam and wrap) are extracted and investigated for the failure of the panel. The stresses in the wrap are higher compared to the other components contributing towards failure of the panel. For pressure analysis, a unit cell cannot be analyzed with periodic BCs. Hence, a quarter panel with six unit cells in the x and y direction was analyzed. A pressure of one atmosphere (1 atm.) is applied on the top face sheet. It was observed that stresses under pressure loads were at least two orders less compared to the thermally induced stresses. Since pressure loads do not cause failure of the panel, they will not be included in the optimization process Buckling Analysis The complex design of the ITPS with face sheets and webs leads to local and global buckling of the laminates under thermal stresses and mechanical stresses. Though 48

49 the webs in the panel are supported by the insulation foam, the buckling of the webs could be due to poor bonding between the foam and the web and also the possibility of damaged foam (due to thermal loads) not being able to support the webs from buckling. The buckling of the ITPS can be carried out on the unit cell or as a large 3-D shell model without any foam (assuming damaged foam). These analyses would provide an estimate of the buckling load. While local buckling, by itself, may not always lead to catastrophic failure, it could contribute indirectly along with high temperatures and stresses. Buckling of the ITPS in ABAQUS is modeled as a eigen value buckling prediction problem where the eigen value can be considered as the load factor at which the structure would fail due to buckling. The analysis is very similar to the stress analysis under thermal loads except for the load step which is linear perturbation procedure. The load considered would be temperature distribution if there is thermal buckling and pressure loading, if there is buckling due to mechanical loads. Figure 3-7. Buckled ITPS due to thermal loads Initially only buckling of the ITPS unit cell without the foam was analyzed (Figure 3-7). The buckling eigen value predicted should be greater than 1.0. Eigen values less than 1.0 suggest that the structure fails at loads lesser than the applied load. For 49

50 instance, if the eigen value is equal to 0.7, it implies that at 70% of the applied load the structure would buckle according to the corresponding buckling mode. If the smallest eigen value is above unity, then the structure will not buckle under the applied loads. Buckling analyses of the ITPS unit cell without foam shows top face sheet buckles which leads to the buckling of the top web. 3.3 Summary The different finite element analyses incorporated in the optimization process is presented in this chapter. Further for more details on the various aspects of the finite element modeling and aforementioned structural analysis, the reader is referred to ABAQUS documentation [77, 78]. Past research included the structural analysis of the ITPS panel with corrugated core [28] using shell elements neglecting the foam. This research considers a complex design with laminates wrapped around a denser foam that cannot be neglected thus requiring 3-D brick elements to model the ITPS panel. As the analysis of the 3-D finite element model is computationally intensive, this research proposes to homogenize the 3-D ITPS panel as an equivalent 2-D orthotropic plate, discussed in detail in Chapter 4. 50

51 CHAPTER 4 FINITE ELEMENT BASED HOMOGENIZATION Structural responses (temperature, stress and buckling load) are computed using high fidelity finite element analyses of the 3-D ITPS panel that are computationally expensive. The complex design of the panel consisting of face sheets, wraps and insulation foams and the fact that these components are made of composite materials make the computational expenses prohibitively large. Moreover, the presence of foam in the panel requires 3-D finite elements to model the panel. A 3-D finite element model (12x12 unit cells) of the ITPS consists of 114,000 nodes and a static thermo-mechanical analysis of the model takes approximately 40 minutes. To reduce the computational cost involved in the analysis of the composite ITPS panel, this research proposes homogenization of a 3-D plate like structure into a 2-D equivalent orthotropic plate. The homogenization method is based on micromechanics approach [79] where a representative volume element (RVE) of the structure is subjected to six linearly independent deformations to predict the equivalent plate stiffness properties. A unit cell is used to estimate the in-plane stiffness properties ([ABD] matrix) and a 1-D plate model is employed to estimate the transverse shear stiffness of the ITPS panel. The finite element model of the ITPS is modeled using 20-node brick elements (C3D20R) in the commercial finite element software ABAQUS. The accuracy of the method is investigated by comparing the responses (displacements and stresses) of the 3-D model of the ITPS panel to the equivalent homogenized plate model. 4.1 A,B,D matrices The unit cell is subjected to six linearly independent deformations to determine the equivalent stiffness properties of the panel. The deformations are applied as periodic displacement boundary conditions (Table 4-1) which includes three mid-plane strains(ε x0,ε y0 and γ xy0 ) and three curvatures (κ x, κ y and κ xy ) [80]. 51

52 Figure 4-1. Schematic view of the ITPS unit cell (RVE) Table 4-1. Periodic boundary conditions for the six deformations [u(a,y) - [v(a,y)- [w(a,y) - [u(x,b) - [v(x,b) - [w(x,b) - u(0,y)] v(0,y)] w(0,y)] u(x,0)] v(x,0)] w(x,0)] ε x = 1 a ε y = b 0 γ xy = 1 0 a/2 0 b/2 0 0 κ x = 1 az 0 -a 2 / κ y = bz -b 2 /2 κ xy = 1 0 az/2 -ay/2 bz/2 0 -bx/2 These deformations create in-plane forces (N x,n y and N xy ) and moments (M x,m y and M xy ) in the unit cell. The constitutive relation between the in-plane force, moments and strains and curvature gives the stiffness properties of the structure [81]. The constitutive relation is given as N x N y N xy M x M y M xy = A 11 A 12 0 B 11 B 12 0 A 12 A 22 0 B 12 B A B 66 B 11 B 12 0 D 11 D 12 0 B 12 B 22 0 D 12 D B D 66 ε x0 ε y0 γ xy0 κ x κ y κ xy (4 1) 52

53 For instance when the unit cell is subjected to in-plane strain ε x0 in the x-direction, the face y z at x= a of the unit cell is displaced by a distance a in the x-direction relative to the face x=0. One of the corner nodes is fixed to prevent rigid body motion. From the nodal forces acting on the nodes on the faces of the unit cell, the in-plane forces and moments can be calculated as follows: N x = 1 b N y = 1 a N xy = 1 b M x = 1 b M y = 1 a M xy = 1 b nn i=1 nn i=1 nn i=1 nn i=1 nn i=1 nn i=1 F (i) x (a, y, z) F (i) y (x, b, z) F (i) y (a, y, z) zf (i) x (a, y, z) zf (i) y (x, b, z) zf (i) y (a, y, z) The in-plane forces would provide the first column of the in-plane stiffness matrix [A] and the moments would give the first column of the coupling stiffness [B] (Equation 4 1). The above procedure is repeated with the other mid-plane strains and curvatures to populate the entire [ABD] matrix. When this procedure is implemented correctly, the calculated [ABD] matrix should be a symmetric matrix. Figure 4-3 shows the deformation of the unit cell for each case of unit strain applied. Figure 4-3a shows how the unit cell would deform due to the periodic boundary conditions applied in Figure 4-2. These figures are intended to show how the unit cells deform due to the periodic boundary conditions. 4.2 Transverse Shear Stiffness In the current design of the ITPS panel, the insulation bars act as the core of the sandwich panel and the wraps act as the shear webs providing considerable shear stiffness. The unit cell model with periodic BCs cannot be used for calculating the transverse shear stiffness. When a transverse shear force, say, Q x is present, it gives (4 2) 53

54 Figure 4-2. Periodic boundary condition applied on the unit cell for ϵ x = 1 Figure 4-3. Schematic view of the unit cell deformations due to strains and curvatures rise to the bending moment M x which varies linearly along the x-direction. Thus the conditions for periodicity are violated and one cannot come up with boundary conditions 54

55 that reflect this variation exactly. Hence, an alternate method with an 1-D ITPS plate model is considered. This method combines 3D finite element analysis and beam/plate theory to calculate the transverse shear stiffness. While the former is accurate, it carries with it the effects of the boundary conditions on both ends of the model. When the model is long the end effects will become insignificant. The beam theory is good for long panels. These two factors together require long beam/1d plate for the present ITPS model. For instance, to determine the transverse shear stiffness A 55 the 1-D plate model with unit cells in the x-direction is considered (Figure 4-4). Figure D ITPS model with unit cells in x- direction The boundary conditions on the 1-D ITPS model are at x = 0, u(0, y, z)=0, Bottom edge w(0, y, 0) =0; at x=l, u(l,, y, z)=0, w(l, y, z) w(0, y, z) = C(C = Constant). Figure 4-4 shows the displacement boundary conditions and plane strain boundary conditions applied on the x z plane [82]. The aim of the analysis is to deflect the 1-D plate model vertically (transverse direction) using a tip force F. The total deflection due to bending and shear deformation can be calculated analytically as 55

56 w(x) = F x A 55 F 2D 11 ( x 3 3 Lx2 2 ) (4 3) D 11 = D 11 B2 11 A 11 (4 4) The first term is shear deformation and the second term is bending deformation. F is the net vertical force due to the displacement boundary condition, L is the length of the 1-D panel and D 11 is the reduced bending stiffness [82]. The bending deflection can be calculated analytically using tip force F and the [ABD] matrix that is already determined. The total deflection can be calculated using finite element analysis and thus the shear stiffness can be calculated using Equation 4 3. Transverse shear stiffness is a cross sectional property and for a homogenous structure which can be considered to be made of infinite number of unit cells, the transverse shear stiffness is independent of the length. However, the ITPS panel considered here has finite number of unit cells. The shear stiffness is overestimated due to the boundary effects. Because of the limited computational resources, it is not affordable to find the shear stiffness using 1-D finite element plate models consisting of large number of unit cells, hence the stiffness value is computed for smaller lengths, 10, 20 and 40 unit cells and then a convergence criterion is applied to find the converged value of the stiffness value. The convergence rate can be expressed as k 3 k 2 k 2 k 1 = { N1 N 2 } α (4 5) where k 1, k 2, k 3 are the values of stiffness for say n = 10, 20 and 40 respectively. N 1 /N 2 is the ratio of the length of the panel (ratio of number of unit cells, here N 1 /N 2 = 10/20 = 20/40 = 0.5). α is the convergence factor which should be greater than unity for convergence of the stiffness. Calculating the convergence factor from n=10,20 and 40, then using Equation 4 5, the stiffness could be predicted for longer panels to obtain a 56

57 converged value of the stiffness property. Similar procedure is carried out for estimation of A 44. The converged transverse shear stiffness along with the [ABD] matrix is used to estimate the accuracy of the homogenized model Deflection Comparison 4.3 Accuracy of the Homogenization Method The ITPS panel is subjected to various combinations of loads when installed on the exterior of the space vehicle. Pressure, temperature, in-plane inertial loads and impact loads are the various loads that act on the panel during flight. In this chapter, pressure loads are applied to compare the response (displacement and stresses) of the 3-D ITPS panel to that of the homogenized plate to evaluate the accuracy of homogenization procedure. Since it is computationally expensive to construct a finite element model of the 3-D ITPS panel of increased lengths (Figure 3-1), an 1-D model consisting of series of unit cells in either (x or y) direction is considered. The 3-D ITPS model is fixed at one end and free at the other end with plane strain boundary conditions parallel to the x-z plane. It is subjected to a pressure load p =1000 Pa (Figure (4-5)). For an equivalent1d orthotropic plate with similar boundary conditions and load, a closed form solution of the deflection is available as shown in Equation 4 6. w(x) = p A 55 ) (Lx x2 2 p 2D 11 ( L 2 x 2 2 ) + x4 12 Lx3 3 The homogenized and the 3-D ITPS model (Figure (4-5)) deflections are compared to evaluate the accuracy of the equivalent stiffness properties estimated Stress Comparison The stresses are compared between the equivalent homogenized plate and ITPS panel by employing the reverse homogenization procedure. The original ITPS model is subjected to pressure load and stresses at various cross sections are determined. For (4 6) the homogenized model subjected to pressure load, at any distance x, mid-plane strains and curvatures can be calculated as 57

58 Figure D model of the ITPS with unit cells in x- direction subjected to pressure load and fixed BCs M x = p(l x)2 2 (4 7) κ x = M x 1 D 11 (4 8) ε x0 = B 11 A 11 κ x (4 9) The mid-plane strains and curvatures of the homogenized plate are applied to the ITPS unit cell using the periodic boundary conditions (Table 4-1) and the stresses are determined. This is referred as reverse homogenizing the equivalent homogenized plate to the 3-D ITPS model. The stresses in the unit cell are compared to that of the ITPS at various points along its length. 4.4 Results and Discussion A,B,D matrices - Equivalent material vs. Equivalent structure Some researchers have considered homogenizing the sandwich panel as a plate made of a homogeneous material. This will be applicable when the panel is subjected to only in-plane forces. However, when there are transverse forces due to surface pressure 58