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Many approximation concepts have been developed to reduce the cost of optimization procedures [36,63]. The methods listed in Reference [36] are sequential approximate optimization [38], global reduced-basis approximations [39], variable-complexity modeling technique [40–41] and response surface (RS) approximation technique [42–47]. The first three techniques use a mixture of simple analytical formulas and complicated analyses. The simple formulas allow a quick move towards the optimum, while the complicated analyses help in “course correction” during the optimization procedure. Although these methods reduce the cost of optimization compared to DSS, other costs are added such as development of an algorithm to effectively combine the simple and complicated analyses, because, such an algorithm would most probably be problem specific and, therefore, general software packages may not be available. Surrogate-based multi-fidelity methods for multi-objective optimization are discussed in Reference [64].

The response surface (RS) technique is a much simpler process wherein RS approximations are used to represent the objective and constraint functions (usually) in the form of polynomials. The polynomials are obtained by fitting design variables to data computed at a set of carefully chosen design points. More advantages and examples of usage of the RS technique are presented later in the section.

Reference [37] also summarizes MDO efforts in aeroelastic optimization but concentrates on research work using high fidelity methods. Typical examples of high-fidelity methods are finite element methods for structural analysis and finite difference or finite volume approaches for solving the Euler/Navier-Stokes equations. Research efforts in optimization using highfidelity methods are classified into 3 categories. The first 2 categories deal with procedures that are characterized by multidisciplinary coupling at analytical and sensitivity levels. The third category is one of uncoupled analysis and is of interest for the ITPS design optimization. The optimization procedure is uncoupled at the analytical level by means of RS approximations.

Once the RS approximations are obtained, the coupling takes place during the optimization procedure. This coupling is analytical in nature and is very inexpensive [37].

Some of the advantages associated with RS approximations technique, drawn from

References [36] and [37] are listed below, along with their relevance to the ITPS design:

1. Each analysis deals with a single discipline at the analysis level, which is usually the most computationally intensive stage of the MDO process. This does not particularly apply to the optimization procedure used for ITPS design as there are no distinct thermal or structural design variables.

2. The RS approximation “smooth out numerical noise” [36,37,42], which usually mislead the optimizer during gradient search. Thus, the derivative based optimizers can converge rapidly to the optimum. For example, the smallest buckling Eigen value can be a very unyieldy response as the first buckling mode may occur in a completely different region of the design space depending on the combination of design variables. Thus, there could be huge kinks in this response, which can lead to increase in the number of iterations or, in worst case, failure to find the optimum value.

3. RS approximations act as global approximations because they typically span over whole or a large part of the design space. These approximations “permit visualizations of the entire design space” [36]. In the case of ITPS design, the RS approximations span over the whole design space.

4. Once the RS approximations are generated, the optimization procedure becomes very inexpensive. The availability of objective function and constraints in the form of polynomials makes it very easy for the optimizer to carry out sensitivity analyses and determine the search directions. In the ITPS design, the MATLAB optimizer, fmincon, can rapidly converge to the optimal solution (if it exists) and it does so in less than 10 iterations.

5. Due to the global nature of the RS approximations, they can be used repeatedly for design studies with multi-objective optimization. It is also very easy to change the optimization parameters to gradually build a problem database.

RS approximation technique became popular in the mid 1990s and since then it has been used in various MDO problems. Some of the applications are mentioned here to illustrate the wide variety of fields in which RS approximation technique has been utilized [48–61]. Some of these are discussed here. As is the case with most MDO problems, this technique was applied for aeroelasticity problems [50] and related aerospace optimization designs [49,51,54]. RS approximations were used to smooth out noise in approximating the range and cruise drag on a high-speed civil transport for different aircraft designs [48]. Quadratic polynomial RS approximations were used for the design of aircraft engine turbine blade component wherein heat transfer, natural frequency and structural stress were taken into account [53]. Automotive crashworthiness design is another field where the RS approximations were used [55,56] to optimize various factors such as impact forces on the passengers, vibration characteristics of the vehicle and general crashworthiness. RS approximations were used for a heavy duty tire design to optimize for flexibility and durability [57]. Polynomials were used to approximate the responses of mass and stiffness of the tires, and strain in the tire.

A major disadvantage associated with RS approximations is that they are very expensive to generate when the number of design variables is large. The larger the number of variables, the larger is the number of design points required to create the approximations. However, in the case of ITPS design, the number of variables varies from 5 to 8 and the number of analyses required to generate sufficiently accurate approximations is manageable. Thus, RS approximations technique is the most suitable method for ITPS design optimization which requires high fidelity finite element heat transfer, buckling, stress and deflections analyses.

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This chapter presents a description of the finite element models and analyses for heat transfer, stress analysis and buckling analysis of the ITPS panels. These FE analyses will be used for generation of response surface approximations.

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Finite element heat transfer analysis helps determine the peak bottom face sheet temperature, which is required to impose the temperature constraint in the ITPS design optimization, and provides the temperature distribution in the panel that can be used for buckling, stress and deflection analyses.

First the choice of incident heat flux profile is explained followed by a discussion on the choice of heat transfer boundary conditions. Then comparison of 1-D and 2-D heat transfer FE models is presented. Finally, the procedure to obtain peak temperatures and temperature distribution is outlined.

3.1.1 Incident Heat Flux and Radiation Equilibrium Temperature Incident heat flux on the vehicle depends on the shape of the vehicle, the trajectories of the ascent and reentry and is completely different for vehicle ascent and reentry conditions. For the design process, incident heat flux of a Space Shuttle-like vehicle is used.

Figure 3-1A shows the heat flux input for ascent conditions, while Figures 3-1B and 3-1C show the heat flux during reentry on the windward and leeward centerlines of the vehicle, respectively, obtained from Reference [9]. The heating profiles for windward and leeward sides are similar during ascent [9]. Windward centerline is a line drawn on the vehicle surface connecting the nose to tail on the bottom side of the vehicle, while leeward centerline joins the nose and tail on the top side. Different curves on each figure are for heating rates at different points along the centerlines. The distance x, on the charts, indicates the distance from the tip of the nose of the vehicle. The nose-tip of the vehicle corresponds to x = 0.0 inches. The heating rates are extremely high for points closer to the nose of the vehicle. Comparing Figures 3-1A and 3-1B, it is evident 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 for heat transfer FE analysis for the design process.

Radiation equilibrium temperature is the temperature of a surface at which the amount of heat radiated by the surface is equal to the amount of incident heat flux onto the surface. The radiation equation of a surface radiating heat to the ambient can be written as,

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where qrad is the radiated heat flux per unit area of the radiating surface, ε is the emissivity of the surface, σ is the Stefan-Boltzmann constant (σ = 5.67 × 10-8 W/m2-K4), Tsurf is the temperature of the radiating surface, and Tamb is the temperature of the ambient. When Tsurf is equal to the radiation equilibrium temperature of the surface, TEq, then qrad will be equal to the incident heat flux, qin, on the surface.

The top surface temperature of a TPS will be close to the radiation equilibrium temperature when heat is input onto the surface. Thus the radiation equilibrium temperature is the primary parameter in choosing the material for the top surface of the ITPS. Only those materials that have a service temperature higher than the radiation equilibrium temperature can be used on the top surface. The peak radiation equilibrium temperatures for reentry heat fluxes (Figure 3-1B) are shown in Table 3-1.

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A typical heating rate used for design is shown in Figure 3-2, which corresponds to a point, x = 827 inches [8]. Peak radiation equilibrium temperature for this heating profile is 968 K. This heating rate was chosen for the design because it would allow the use of some of the frequently used high temperature metallic alloys like titanium alloys and Inconel.

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Figure 3-2. Typical heating profile used for the ITPS design.

3.1.2 Loads, Boundary Conditions and Assumptions Loads and boundary conditions for the heat transfer problem are schematically illustrated in Figure 3-3. Initial temperature of the structure is assumed as 295 K (72 F). Heat flux is incident on the top surface of the top face sheet. 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. The bottom surface of the bottom face sheet is 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. The optimization with such an assumption would lead to conservative designs. It is also assumed that there is no lateral heat flow out of the unit cell, that is, the heat flux incident on a unit cell is completely absorbed by that unit cell only. In an actual ITPS panel, heat would flow into the stringers and frames 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. Thus this assumption could also lead to a conservative design as the amount of thermal mass is being reduced due to non-inclusion of stringers and frames in the preliminary design. However, the temperature distribution could be completely different in case of heat flowing out of the unit cell. This is not taken into account in the design.

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Figure 3-3. Schematic of loading and boundary conditions for the heat transfer problem.

Ambient temperatures for the top surface of top face sheet are assumed to be 213 K for initial reentry period (0 to 450 seconds), 243 K for second reentry phase (450 to 1575 seconds) and 273 K for final reentry phase (1575 to 2175 seconds). The ambient temperatures are assumed values due to non-availability of this data. Initial heat transfer analyses indicate that the peak bottom face sheet temperature occurs after vehicle touchdown [8]. Therefore, after touchdown the FE analysis continues for another 50 minutes 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 the top surface to simulate the heat transfer to the surroundings while the vehicle is standing on the runway. The value of convective heat transfer coefficient, h, used was 6.5 W/m2-K (6.94×10-4 Btu/s-ft2-°R) [8]. The ambient temperature during this period was assumed to be 295 K.

The transient heat transfer analysis is divided into 4 load steps as shown in Table 3-2.

These steps were chosen to approximately represent a typical Space Shuttle-like heat flux input on the top surface as shown in Figure 3-2. Step 1 simulates the initial reentry period, when the heating rate is linearly ramped up from 0 to 3.0 Btu/ft2.sec (34,069 W/m2), Step 2 simulates the second phase of reentry, when the heating rate is linearly ramped up from 3.0 to 3.5 Btu/ft2.sec (34069 to 39748 W/m2), and Step 3 simulates the final reentry phase, when the heating rate is ramped down from 3.5 to 0.0 Btu/ft2.sec. During these load steps, the top surface has incident heat flux loading and radiation boundary conditions. During Step 4 (vehicle sitting on the runway) there is no heat input and the top surface has radiation and convective heat transfer boundary conditions.

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