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K0 is the stiffness matrix corresponding to the base state which includes the preloads, K∆ is the differential initial stress and load stiffness matrix due to the incremental loading pattern (Q), λi are the eigenvalues, ui are the buckling mode shapes or eigenvectors, where the subscript i refers to the ith buckling mode. Solving the eigenvalue problem Equation (3.6) gives the eigenvalues and the eigenvectors. The critical buckling load for each buckling mode will then be the sum of preloads plus the scaled incremental load, P + λiQ. If the eigenvalue is equal to 0.7, then it implies that at 70% of the applied load the structure would buckle according to the corresponding buckling mode. If the smallest eigenvalue is above unity, then the structure will not buckle under the applied loads. In case of the ITPS design, all loads are critical and therefore the value of P is equal to zero.

For the ITPS buckling problem, the FE analysis was divided into two steps. Step 1 corresponds to the base state. In this state all the boundary conditions were imposed. In Step 2 (eigenvalue step), temperature, pressure and in-plane loads were imposed.

In ABAQUS buckling analysis, temperature dependent material properties are not taken into account in the buckling step. The material properties imposed in the base step are “carried over” into the buckling step. The base state temperature in buckling analysis is equal to the initial stress free temperature, typically 295 K. Thus the material properties used in the buckling steps are the properties corresponding to this temperature. Even if additional nodal temperatures are imposed in the buckling step, the properties used for eigenvalue calculations will be the same as those corresponding to the initial temperature. Taking into account the range of temperatures that an ITPS is subjected to, this will clearly not yield an accurate solution as the material properties can vary to a large extent with temperature.

A method was designed to approximately impose the temperature dependent material properties on the ITPS structure. As mentioned earlier in this section, the temperature distribution on the ITPS panel is known in advance from the heat transfer analysis (cubic polynomial in z-coordinate). From this the top and bottom face sheet temperatures were obtained and the material properties corresponding to this temperature were imposed on the face sheets. In case of webs, however, the temperature varies through the thickness of the panel. Therefore, each web was divided into 10 partitions from top to bottom as shown in Figure 3-10. The average temperature in each partition was obtained from the temperature distribution data. Material properties corresponding to the average temperature of each partition were assigned to that particular partition. It should be noted that since the temperature distribution is a function of zcoordinate only, all partitions with the same z-coordinate will be assigned the same material properties.

In ABAQUS buckling eigenvalue problem, the desired number of eigenvalues can be specified. For example, if 15 eigenvalues are desired, then ABAQUS extracts the first 15 smallest eigenvalues and the corresponding eigenvectors. There may be many negative eigenvalues in the first 15 values. In order to avoid negative eigenvalues, a minimum eigenvalue of interest may be specified. Typically, a small positive value, example 10-6, is specified, so that all the 15 eigenvalues obtained from the analysis are positive.

Figure 3-10. Typical ITPS panel illustrating the manner in which the webs are partitioned into 10 regions to impose approximate temperature dependent material properties.

The output data of interest from the buckling analysis are the smallest eigenvalues. These values can be obtained from the ABAQUS data files, which have the extension ‘.dat’. The eigen vector corresponding to each eigenvalue can be obtained from the output file in the form of nodal displacements. The nodal displacement data for buckling problems are normalized so that the maximum nodal displacement is always equal to unity. By obtaining the node number whose displacement is equal to unity and combining this with the nodal coordinates data, the buckling position can be identified. For example if the node with maximum displacement is in the top face sheet, then the buckling eigenvalue corresponds to top face sheet buckling. The buckling modes can also be visualized in the ABAQUS visualization module window. Typical buckling modes are shown in Figure 3-11. The position of buckling can be useful when the buckling response surface approximations are obtained separately for web buckling and top face sheet buckling, for example (more details in Chapter 5).

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Stress analysis of the ITPS panel is carried out using the same geometry and mesh as the buckling analysis. The buckling step is replaced by static stress analysis step. In stress analysis case, ABAQUS is capable of taking into account the temperature dependent material properties.

Therefore, even though the web is partitioned into 10 regions, all the regions are give the same temperature dependent material properties as the web material. The ABAQUS solver automatically assigns the material properties to each element according to the temperature imposed on the element.

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Figure 3-11. Typical buckling modes of the ITPS panel. A) Web buckling. B) Top face sheet buckling.

From the stress analysis models, deflections and von Mises stress values can be obtained.

The z-direction deflection of the top face sheet can be obtained separately from the ABAQUS output files. Similarly, the von Mises stresses for each section of the ITPS can be obtained separately.

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This chapter explains the process for generation of response surface approximations used to approximately represent the constraints of the optimization problem. Then the optimization procedure is presented.

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For the current ITPS design problem, a quadratic response surface approximation of each constraint was adopted to solve the optimization problem. These response surface (RS) approximations are functions of the design variables. The procedure for obtaining these RS approximations is outlined below.

A quadratic RS approximation in 7 variables has 36 coefficients and hence at least 36 function evaluations are required to determine all the coefficients. Usually, the number of function evaluations, N, required for obtaining a sufficiently accurate approximation is twice the minimum number of coefficients and depends upon the nature of the problem at hand. For ITPS design, N has been typically found to be greater than 150 in order to obtain sufficiently accurate approximations. The major reason for this unusually high value of N is the RS approximation for buckling eigenvalues. RS approximation for the peak bottom face sheet temperature has been found to be sufficiently accurate even for N equal to 100. However, the buckling eigenvalue response has large kinks, which are difficult to capture using a quadratic polynomial function. As a result the number of responses has to be increased in order to get an accurate polynomial.

An RS approximation obtained using least squares approximation method cannot be guaranteed to be accurate if a variable value is chosen outside its range. To avoid ill-conditioned matrices while implementing the Least Squares Approximation technique, normalized values of the variable should be used so that each variable varies between zero and unity.

Optimized Latin Hypercube Sampling (LHS)technique was used for the design of experiments to obtain N different combinations of the design variables, also referred to as design points. In the LHS technique the specified range of each variable is divided into N equal intervals. Then, one value from each interval for each variable is randomly chosen. These values are randomly combined to generate N design points. The optimized LHS technique performs iterations to maximize the minimum distance between two values of a variable. This scheme precludes the possibility of the variables being clustered in any region of the design space and ensures uniform distributions of the points in the design space.

4.1.1 Response Surface Approximations for Maximum Bottom Face Sheet Temperature Transient 1-D heat transfer analysis was carried out for each of the N design points using ABAQUS. As described in Section 3.1.5, the peak bottom face sheet temperature was obtained for each of these design points and the coefficients of the RS approximation were obtained by least squares approximation. RS approximation for peak bottom face sheet temperature is usually quite accurate. More discussion about accuracy is given later in Chapter 5.

4.1.2 Response Surface Approximations for Buckling For buckling problem the RS approximations were obtained for the smallest buckling eigenvalue. The same design of experiments used for heat transfer analysis was used for the buckling analysis. Using the combination of variables obtained by Optimized LH Sampling, the 3-D FE model for buckling was created and analyzed as described in Chapter 3. The first 15 positive eigenvalues were obtained for each design point. The buckling position for each of these eigenvalues was then identified. From this data the smallest eigen buckling value for top face sheet, web and bottom face sheet were obtained. Using these eigenvalues and the normalized values of the design variables, RS approximations can be obtained for the smallest buckling eigenvalue in each section separately.

For some design points, the buckling value for a particular section does not figure in the first 15 values. For example, if the total number of design points is 200, then web buckling may figure in the first 15 values for only 180 of these design points. In that case, only these 180 points will be considered for the generation of the web buckling RS approximation. If the number of design points available for RS approximation generation is less than the minimum required for the least squares approximation, then that particular constraint will not be considered for the optimization procedure. This can be justified by stating that for the range of design variables chosen for this optimization, buckling of that particular section will not be an influential factor. However, if that particular constraint proves to be critical at the optimum design, then the variable bounds need to be suitable altered so as to include sufficient number of design points at which this constraint generates a response.

4.1.3 Response Surface Approximations for Stress and Deflection RS approximations for stresses were obtained by extracting the maximum von Mises stresses for each design point. Maximum stresses were obtained separately for each of the 3 sections - top face sheet, bottom face sheet and web. For each of these (shell) sections, von Mises stresses were obtained at the top surface and bottom surface separately and the higher value among the two was considered for the generation of the RS approximations.

The maximum and minimum z-direction displacements are obtained separately for the top face sheet and the absolute value of the difference is used for the deflection generation of RS approximations.

4.2 Procedure for Generation of Response Surfaces Approximations Typically, it has been observed that more than 150 function evaluations are required to generate sufficiently accurate response surface approximations for 6 design variables. This would imply that a large number of FE experiments need to be carried out for one optimization.

Further, the process has to be repeated whenever there is a change in the design geometry or other parameters like heat load, pressure load and boundary conditions. It is a formidable number of FE analyses to be carried out manually.

A Matlab® code has been developed for this purpose called the ITPS Optimizer. The ITPS optimizer activates ABAQUS® and performs the FE heat transfer, buckling and stress analyses automatically. The functions of ITPS Optimizer are illustrated by the flow chart in Figure 4-1.

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Figure 4-1. Flowchart illustrating the procedure followed by the ITPS Optimizer. Rounded boxes represent inputs for FE analyses, rectangles for FE analyses and ellispses represent FE output. Temperature distributions are output of heat transfer analysis and input for buckling and linear stress analysis. The dotted rectangle containing the ellipses represents the data output from ITPS Optimizer, which is used for generating response surface approximations.

The ITPS Optimizer first obtains a design of experiments using Latin-Hypercube Sampling technique. In Matlab®, the function lhsdesign( ) can be used for this purpose. A statement of the form NV = lhsdesign(NExpts, NVars,'criterion', 'maximin', 'iterations',NIter) will assign a matrix of NExpts rows and NVars columns to the matrix NV. NExpts is the number of experiments and NVars stands for the number of variables. The last four commands in the parenthesis are for maximizing the minimum distance between the variable values by iterating NIter times, typically 200. This will ensure that the variables are properly spaced in their respective ranges. The matrix NV will have values between 0 and 1 only. These values may be viewed as normalized variables, normalized by the ranges of the respective intervals of the variables. They are scaled to obtain the exact variable values matrix V. Each row of V represents one combination of the variables or one design point. Using the NExpts combinations of variables the FE experiments are carried out.

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