An important part of an aerodynamic design process is optimizing designs to maximize quantities such as lift and the lift-to-drag ratio, in a process known as shape optimization. It is the goal of this thesis to develop and apply understanding of mixed finite element method and sensitivity analysis in a way that sets the foundation for shape optimization. The open-source Incompressible Flow Iterative Solution Software (IFISS) mixed finite element method toolbox for MATLAB developed by Silvester, Elman, and Ramage is used. Meshes are produced for a backward-facing step problem, using built-in tools from IFISS as well as the mesh generation software Gmsh, and grid convergence studies are performed for both sets of meshes along a sampled data line to ensure that the simulations converge asymptotically with increasing mesh resolution. As a preliminary study of sensitivity analysis, analytic sensitivities of velocity components along the backward-facing step data line to inflow velocity parameters are determined and verified using finite difference and complex step sensitivity values. The method is then applied to pressure drag calculated by integrating the pressure over the surface of a circular cylinder in a freestream flow, and verified and validated using published simulation data and experimental data. The sensitivity analysis study is extended to shape optimization, wherein the shape of a circular cylinder is altered and the sensitivities of the pressure drag coefficient to the changes in the cylinder shape are determined and verified. / Master of Science / When looking at designing an aircraft, it is important to consider the forces air flow exerts on the wings. The primary forces of interest for aerodynamic analysis are lift, which generally acts upward perpendicular to the flow of air, and drag, which opposes the motion of the wing through the air. Optimization is the process of developing a design in such a way that a specific quantity, such as lift or drag, is either maximized or minimized. Many methods exist of predicting the behavior of air flow, and various methods of optimization exist which take already existing predictive software and progressively alter the design to try to meet the minimized or maximized objective. This thesis outlines a multi-step effort to modify an open source software such that it could be used for design optimization.

In this dissertation, a continuum sensitivity method is developed for efficient and accurate computation of design derivatives for nonlinear aeroelastic structures subject to transient<br />aerodynamic loads. The continuum sensitivity equations (CSE) are a set of linear partial<br />differential equations (PDEs) obtained by differentiating the original governing equations of<br />the physical system. The linear CSEs may be solved by using the same numerical method<br />used for the original analysis problem. The material (total) derivative, the local (partial)<br />derivative, and their relationship is introduced for shape sensitivity analysis. The CSEs are<br />often posed in terms of local derivatives (local form) for fluid applications and in terms of total<br />derivatives (total form) for structural applications. The local form CSE avoids computing<br />mesh sensitivity throughout the domain, as required by discrete analytic sensitivity methods.<br />The application of local form CSEs to built-up structures is investigated. The difficulty<br />of implementing local form CSEs for built-up structures due to the discontinuity of local<br />sensitivity variables is pointed out and a special treatment is introduced. The application<br />of the local form and the total form CSE methods to aeroelastic problems are compared.<br />Their advantages and disadvantages are discussed, based on their derivations, efficiency,<br />and accuracy. Under certain conditions, the total form continuum method is shown to be<br />equivalent to the analytic discrete method, after discretization, for systems governed by a<br />general second-order PDE. The advantage of the continuum sensitivity method is that less<br />information of the source code of the analysis solver is required. Verification examples are<br />solved for shape sensitivity of elastic, fluid and aeroelastic problems. / Ph. D.

Novel aircraft configurations tend to be sized by physical phenomena that are largely neglected during conventional fixed wing aircraft design. High-fidelity fluid-structure interaction that accurately models geometric nonlinerity during a transient aeroelastic gust response is critical for sizing the aircraft configuration early in the design process. The primary motivation of this research is to develop a continuum shape sensitivity method that can support gradient-based design optimization of practical and multidisciplinary high-fidelity analyses. A local continuum sensitivity analysis (CSA) that utilizes spatial gradient reconstruction (SGR) and avoids mesh sensitivities is presented for shape design derivative calculations. Current design sensitivity analysis (DSA) methods have shortcomings regarding accuracy, efficiency, and ease of implementation. The local CSA method with SGR is a nonintrusive and element agnostic method that can be used with black box analysis tools, making it relatively easy to implement. Furthermore, it overcomes many of the accuracy issues documented in the current literature. The method is developed to compute design derivatives for a variety of applications, including linear and nonlinear static beam bending, linear and nonlinear transient gust analysis of a 2-D beam structure, linear and nonlinear static bending of rectangular plates, linear and nonlinear static bending of a beam-stiffened plate, and two-dimensional potential flow. The analyses are conducted using general purpose codes. For each example the design derivatives are validated with either analytic or finite difference solutions and practical numerical and modeling considerations are discussed. The local continuum shape sensitivity method with spatial gradient reconstruction is an accurate analytic design sensitivity method that is amenable to general purpose codes and black box tools. / Ph. D.

The method of Continuum Sensitivity Analysis (CSA) with Spatial Gradient Reconstruction (SGR) is presented for calculating the sensitivity of fluid, structural, and coupled fluid-structure (aeroelastic) response with respect to shape design parameters. One of the novelties of this work is the derivation of local CSA with SGR for obtaining flow derivatives using finite volume formulation and its nonintrusive implementation (i.e. without accessing the analysis source code). Examples of a NACA0012 airfoil and a lid-driven cavity highlight the effect of the accuracy of the sensitivity boundary conditions on the flow derivatives. It is shown that the spatial gradients of flow velocities, calculated using SGR, contribute significantly to the sensitivity transpiration boundary condition and affect the accuracy of flow derivatives. The effect of using an inconsistent flow solution and Jacobian matrix during the nonintrusive sensitivity analysis is also studied. Another novel contribution is derivation of a hybrid adjoint formulation of CSA, which enables efficient calculation of design derivatives of a few performance functions with respect to many design variables. This method is demonstrated with applications to 1-D, 2-D and 3-D structural problems. The hybrid adjoint CSA method computes the same values for shape derivatives as direct CSA. Therefore accuracy and convergence properties are the same as for the direct local CSA. Finally, we demonstrate implementation of CSA for computing aeroelastic response shape derivatives. We derive the sensitivity equations for the structural and fluid systems, identify the sources of the coupling between the structural and fluid derivatives, and implement CSA nonintrusively to obtain the aeroelastic response derivatives. Particularly for the example of a flexible airfoil, the interface that separates the fluid and structural domains is chosen to be flexible. This leads to coupling terms in the sensitivity analysis which are highlighted. The integration of the geometric sensitivity with the aeroelastic response for obtaining shape derivatives using CSA is demonstrated. / Ph. D. / Many natural and man-made systems exhibit behavior which is a combination of the structural elastic response, such as bending or twisting, and aerodynamic or fluid response, such as pressure; for example, flow of blood in arteries, flapping of a bird’s wings, fluttering of a flag, and flight of a hot-air balloon. Such a coupled fluid-structure response is defined as aeroelastic response. Flight of an aircraft through turbulent weather is another example of an aeroelastic response. In this work, a novel method is proposed for calculating the sensitivity of an aircraft’s aeroelastic response to changes in the shape of the aircraft. These sensitivities are numbers that indicate how sensitive the aircraft’s responses are to changes in the shape of the aircraft. Such sensitivities are essential for aircraft design. The method presented in this work is called Continuum Sensitivity Analysis (CSA). The main goal is to accurately and efficiently calculate the sensitivities which are used by optimization tools to compute the best aircraft shape that suits the customers needs. The key advantages of CSA, as compared to the other methods, are that it is more efficient and it can be used effectively with commercially available (nonintrusive) tools. A unique contribution is that the proposed method can be used to calculate sensitivities with respect to a few or many shape design variables, without much effort. Integration of structural and fluid sensitivities is carried out first by applying CSA individually for structural and fluid systems, followed by connecting these together to obtain the coupled aeroelastic sensitivity. We present the first application of local formulation of CSA for nonintrusive implementation of high-fidelity aeroelastic sensitivities. The following challenging tasks are tackled in this research: (a) deriving the sensitivity equations and boundary conditions, (b) developing and linking computer codes written in different languages (C++, MATLAB, FORTRAN) for solving these equations, and (c) implementing CSA using commercially available tools such as NASTRAN, FLUENT, and SU2. CSA can improve the design process of complex aircraft and spacecraft. Owing to its modularity, CSA is also applicable to multidisciplinary areas such as biomedical, automotive, ocean engineering, space science, etc.

In this thesis, a continuum sensitivity analysis method is presented for calculation of shape sensitivities of an elastic bar. The governing differential equations and boundary conditions for the elastic bar are differentiated with respect to the shape design parameter to derive the continuum sensitivity equations. The continuum sensitivity equations are linear ordinary differential equations in terms of local or material shape design derivatives, otherwise known as shape sensitivities. One of the novelties of this work is the derivation of three variational formulations for obtaining shape sensitivities, one in terms of the local sensitivity and two in terms of the material sensitivity. These derivations involve evaluating (a) the variational form of the continuum sensitivity equations, or (b) the sensitivity of the variational form of the analysis equations. We demonstrate their implementation for various combinations of design velocity and global basis functions. These variational formulations are further solved using finite element analysis. The order of convergence of each variational formulation is determined by comparing the sensitivity solutions with the exact solutions for analytical test cases. This research focusses on 1-D structural equations. In future work, the three variational formulations can be derived for 2-D and 3-D structural and fluid domains. / Master of Science / When solving an optimization problem, the extreme value of the performance metric of interest is calculated by tuning the values of the design variables. Some optimization problems involve shape change as one of the design variables. Change in shape leads to change in the boundary locations. This leads to a change in the domain definition and the boundary conditions. We consider a 1-D structural element, an elastic bar, for this study. Subsequently, we demonstrate a method for calculating the sensitivity of solution (e.g. displacement at a point) to change in the shape (length for 1-D case) of the elastic bar. These sensitivities, known as shape sensitivities, are critical for design optimization problems. We make use of continuum analytical shape sensitivity analysis to derive three variational formulations to compute these shape sensitivities. The accuracy and convergence of solutions is verified using a finite element analysis code. In future, the approach can be extended to multi-dimensional structural and fluid domain problems.