# Research

## Research Components

The objective is to develop, implement, and evaluate a novel methodology for automated multidisciplinary optimization of complex aircraft configurations, including a CATIA V5 geometry parameterization interface, a robust mesh movement technique, a parallel Newton-Krylov flow solver, a structural finite-element model, a coupled-adjoint approach, aeroservoelasticity and an optimizer. Many of the issues of current interest in the field of high-fidelity aircraft design optimization will be addressed. Three principal project areas with significant overlaps can be identified:

- Aerodynamic shape optimization
- Control system for flutter suppression
- High-fidelity MDO of aircraft configurations

## Methodology and Mathematical Techniques Involved

### Aerodynamic Shape Optimization

The basis for the proposed research is the Newton-Krylov algorithm developed in Prof. Zingg's group for two-dimensional aerodynamic optimization (the associated computer program is known as Optima2D). It is described as a Newton-Krylov algorithm because 1) the flow solver is based on an inexact-Newton strategy using a preconditioned Krylov method to solve the linear problem at each Newton iteration, 2) the same preconditioned Krylov method is used to solve the discrete adjoint equation used in the calculation of the gradient, and 3) the optimizer is a quasi-Newton method for unconstrained optimization. Constraints are handled through a penalty approach in which constraint violations penalize the objective function. The current two-dimensional geometry parameterization is based on B-splines. Optima2D has proven to be an efficient tool for the design of two-dimensional airfoil sections and multi-element high-lift configurations. This algorithm represents the state of the art in terms of efficiency, reliability, and robustness. The goal of the current projects in this area is to develop a similarly efficient, reliable, and robust capability in three dimensions, which is considerably more difficult.

#### CATIA V5 CAD-based geometry control system

A CAD-based geometry control system is useful when the initial geometry is in CAD, and the final geometry must also be specified in the CAD environment. The difficulty this introduces is that the underlying geometry parameterization is typically proprietary. The geometry cannot be removed from the CAD environment without loss of information; hence the geometry must be manipulated within the CAD environment. We will develop a CAD-based system to parameterize and modify complex geometries.

The geometry parameterization will be built on the parametric and feature-based solid modeling of the CATIA V5 geometry kernel. This will allow the CATIA V5 geometry definition to be interrogated and modified through the CAPRI application programming interface. Perturbing the surface grid in response to a geometry modification will be complex due to the parametric modeling in CATIA V5. To overcome this dynamic geometric representation, an internal representation of the volume will be generated and interrogated after the geometry modification in order to modify the surface grid. The geometry will never leave its native CAD environment, and hence no information will be lost during the optimization cycle. The system will be developed such that it is straightforward to vary conventional aerodynamic parameters, such as sweep angle, taper ratio, etc.

#### Mesh deformation using elasticity

The first approach will build on work completed for the Seed Proposal that used a mesh based on solutions of the linear elasticity problem. To preserve mesh quality, the stiffness used in the linear elasticity problem is high in small cells, and is increased during deformation for cells whose quality is decreasing. This approach has been shown to be quite robust, even for large mesh deformations. We have shown that solving the adjoint problem for this mesh deformation significantly improves the convergence of optimization problems when compared with computing gradients using finite differences. To make this approach truly efficient for optimization, however, will require work to improve the performance of the mesh movement. We have also developed a faster approach in which a coarse supermesh is used to control the volume mesh. The mesh movement algorithm is applied to the supermesh, and a new volume mesh is generated in the interior of the supercells by an interpolation technique based on B-splines. This novel integrated approach to geometry paramaterization and mesh movement has proven to produce high quality meshes in a fraction of the time needed to apply the linear elasticity approach to the volume mesh.

#### Extension of flow solution and gradient computation to turbulent flows

The required level of fidelity of the flow solver necessitates the numerical solution of the Reynolds-averaged Navier-Stokes equations with a turbulence model plus the capability of predicting laminar-turbulent transition. For use in optimization, the solver must converge rapidly and reliably. For full convergence to the optimum, an accurate gradient is needed. The gradient computation should not require substantially greater computational expense than the flow solution. The preferred approach is thus the discrete adjoint method with as accurate a linearization of the discrete residual equations as possible. In order to further reduce the computing times needed for optimization, we will incorporate higher-order methods, which produce accurate solutions on coarser meshes. This will not only speed up the flow solve, but the mesh movement as well. Finally, we will incorporate a transition prediction capability into the flow solver.

The current parallel solver for the Euler equations will be extended to viscous turbulent flows based on the Spalart-Allmaras turbulence model. Our algorithm has several novel features. We utilize simultaneous approximation terms (SAT's) at boundaries and block interfaces. This reduces the mesh continuity requirements across interfaces, reduces the communication needed, thus simplifying parallelization, and simplifies the implementation of higher-order methods. The iterative method is a Newton-Krylov algorithm in which the linear solver is GMRES with ILU preconditioning, parallelized using an approximate Schur technique. With a number of tricks, this approach has proven effective in three dimensions. For example, we obtain a solution to the Euler equations with lift and drag coefficients converged to three figures on a mesh with nine million nodes in 3.5 minutes on 768 processors. The algorithm is scaling well and provides truly state of the art performance. Extension to turbulent flows will require the development of suitable SAT terms for viscous operators. For efficient gradient computation, a three-dimensional parallel adjoint solver will be developed based on a full hand linearization of the solver and a flexible variant of the GCROT linear solver preconditioned using an ILU factorization again parallelized using an approximate Schur technique. We will also extend the new augmented adjoint technique to three dimensions and extend the three-dimensional solver to include the new lift constraint. Our approach to higher-order methods will be based on summation-by parts operators. To date, the Euler solver has been raised to fourth order.

#### Automated strategies for practical design

An automated treatment of multipoint problems is needed for practical design. A practical aerodynamic configuration or component must operate effectively under a broad range of both on- and off-design operating conditions. Off-design conditions are typically treated as constraints. We address such problems by defining a composite objective function consisting of a weighted sum of the objective functions defined for the individual operating conditions. Some of the weights, typically for the on-design conditions, are determined from the priorities of the designer. Others can be chosen automatically. For example, the weight on an off-design condition can be chosen to be as small as possible while satisfying the desired constraint. Automating the weight selection is an important step toward a hands-off approach to optimization. We seek to develop an automated procedure that determines the off-design point weights and produces Pareto fronts to provide the designer with the trade-offs associated with logical groupings of on-design conditions.

We will consider two approaches to the problem defined above. The weights for the on-design points must be specified by the designer. Pareto fronts can be used to gain an understanding of the trade-offs associated with these weights. For the off-design operating conditions, we will iteratively determine the minimum non-negative weight that ensures that the off-design constraint is satisfied. This approach will work with the quasi-Newton method for unconstrained optimization. Alternatively, we will evaluate an approach based on sequential quadratic programming that will permit the off-design operating requirements to be formally handled as constraints. In addition, we will develop a formal technique to ensure that the number of operating points considered is sufficient to avoid the development of specialized optima, i.e., far superior performance at the specified operating points compared with nearby operating conditions. This will be accomplished by periodic sampling at intermediate operating points and addition of supplementary conditions as required.

#### Numerical Optimization

This is not an area where the principal investigators have the expertise needed to make major contributions. Hence we will continue to use methods developed elsewhere, such as quasi-Newton and sequential quadratic programming (SQP) methods. Although we have had success with the quasi-Newton method coupled with a penalty approach to constraints, this methodology does have some shortcomings, which the SQP approach can potentially alleviate. To this end we now use SNOPT for many problems of interest.

This part of the proposed project will integrate the aerodynamic shape optimization with structural optimization to develop the capability to perform aero-structural shape optimization. This requires the extension of the adjoint equations to a coupled system of adjoint equations that poses significant mathematical and computational challenges. In addition, the aeroservoelasticity project will use results from the optimized aero-structural wing to analyze its flutter characteristics and to design an optimal control law for maximum flutter suppression.

### Control System for Flutter Suppression

Aerodynamic flutter refers to a subject that has evolved from the beginning of manned flight. Aeroelasticity and structural dynamics occupy a prime role in the current design of advanced aircraft, missiles, launch vehicles and spacecraft. With the advent of modern flight control systems (FCS), the disciplines of aeroelasticity and structural dynamics are linked together by the servoelastic interaction between the aeronautical structure and the FCS; this is known as aeroservoelasticity (ASE). Aeroservoelasticity is defined as the interaction between structural dynamics, unsteady aerodynamics and the flight control system dynamics.

The servoelastic coupling (also referred to as structural coupling) is caused when sensors attached to the airframe acquire not only the rigid aircraft motions needed for the flight control system inputs, but also the vibratory response of the flexible structure. It is the feedback of the vibration that must be compensated for in order to maintain sufficient structural stability margins over the range of operating conditions for the vehicle. From an airworthiness perspective, ASE interactions can lead to increased dynamic loads on aero-surfaces (ailerons, elevators, etc), cause fatigue cycles on critical flight structures (caused by vortex bursts on vertical tail fins; induced buffeting is an example) as well as have a negative influence on pilot tasks. In more severe cases, ASE interactions can cause limit cycle oscillations in primary flight structures such as a lifting surface.

Much of the work on active flutter suppression has been focused on linear aeroelastic systems without taking into account actuator saturation. However, real flight vehicle structures exhibit nonlinear characteristics, which manifest themselves as limit cycle oscillations. In addition flight vehicle structures are also subject to limits in actuation. For free play nonlinearity, limit cycle motion is dependent on freestream velocity, the initial pitch condition and the size of the free play nonlinearity. The type of nonlinearity that will be studied in the proposed research is a continuous structural nonlinear spring in both bending and torsion. Departing from previous work, the proposed research will not use a linear model. It will instead approximate the linear model by a piecewise-affine model and design switched controllers for flutter suppression.

The objective of the proposed research is to study switched flapping controllers, such as piecewise-affine, gain scheduling and linear parameter varying controllers to suppress flutter and limit cycle oscillations in the presence of actuator saturation. The work will also study the effectiveness of the flapping controllers as a function of the location of the elastic axis. The work will be based initially on a two degree-of-freedom model.

### High-Fidelity MDO of Aircraft Configurations

This part of the proposed project will integrate the aerodynamic shape optimization with structural optimization to develop the capability to perform aero-structural shape optimization. This requires the extension of the adjoint equations to a coupled system of adjoint equations that poses significant mathematical and computational challenges. In addition, the aeroservoelasticity project will use results from the optimized aero-structural wing to analyze its flutter characteristics and to design an optimal control law for maximum flutter suppression.

#### Mesh generation and deformation for structural finite-element models

We currently have the capability of generating meshes for a structural FEM model given the surface of a wing. However, for a more consistent and accurate handling of the geometry, it is necessary to link this mesh deformation to the same CATIA V5 geometry mentioned previously for aerodynamic shape optimization. Changes in the structural mesh will also be applied through the CAPRI application programming interface. This will handle the movement of the FEM nodes located at the surface.

In this research, we will use FEM meshes that are much more refined, and we will allow for much greater freedom in the design changes when compared to our previous work. The major change in the set of shape design variables is the addition of planform variables. This represents a major challenge for handling the geometry (hence the use of CATIA V5), the CFD mesh perturbation, and the structural mesh deformation. Therefore, the structural mesh deformation that controls the nodes that are not on the surface has to be greatly improved in order to preserve accuracy. A scheme that uses the FEM equations of elasticity will be used. The advantage in this case is that these equations are already formed when solving for the displacement using the FEM.

The crucial mathematical component of this project is the sensitivity analysis of this mesh deformation. For accurate and efficient sensitivities, an adjoint method will be used, which will then integrate with the coupled aero-structural adjoint equations mentioned below.

#### Aero-structural coupling

In the last few years, we have developed the capability of performing high-fidelity aero-structural analysis for aircraft configurations. While great care was taken to ensure that the load and displacement transfer preserved the high-fidelity of the CFD and FEM by developing a consistent and conservative scheme, there are areas where this scheme can be improved.

The first has to do with the displacement transfer, which does not necessarily preserve C1 continuity of the deformed fluid-structure interface. Thus, we will develop a displacement transfer scheme that preserves the smoothness of the surface, in a way that is consistent with the assumptions of the plate finite element that is used to model that same surface. This will take into consideration the particular shape functions of the finite element, as well as the constitutive matrix of the material.

The other aspect is related to the fact that both the FEM analysis and the load-displacement transfers are linear. While this is a reasonable model for many aircraft configurations under certain load conditions, it yields inaccurate results for flexible wings with high aspect ratios, which have emerged in many recent unmanned aerial vehicle designs. To analyze such configurations, we will develop a Newton algorithm with an arc-length continuation method that can efficiently handle nonlinear structures.

#### Coupled adjoint sensitivities

The coupled-adjoint equations and their solution will be one of the most interesting projects from a mathematical point of view. Previous work has been limited by the high computational cost of forming the off-diagonal terms in the Jacobian of the coupled system. These terms represent the sensitivities of the governing equation residuals of a discipline with respect to the state variables of another discipline. Thus far, we have computed these terms using finite differences. In the proposed project, we will adopt a more efficient approach that uses automatic differentiation selectively, in a similar fashion to what we have done for CFD solvers. The difference here is that the boundaries between the two solvers (CFD and structural FEM in this case) must remain clearly defined throughout the differentiation process.

Since the off-diagonal block matrices are usually very sparse, we also have the opportunity to exploit this fact to develop an even more efficient coupled-adjoint algorithm. Once this new scheme is in place, and depending on the performance we achieve, we will consider the trade-off between accuracy and speed. It is possible to neglect less significant terms, converge the solution to a larger tolerance, or both.

#### MDO of aircraft configurations

Once a scheme is in place for accurate and efficient computation of total coupled sensitivities, we will be able to perform high-fidelity MDO of aircraft configurations. The sensitivities of crucial functionals, such as drag coefficient, structural weight and structural stresses, will be computed with respect to the design variables. These variables will include shape design variables describing the wing and fuselage surfaces that will be controlled through CAPRI.

The proposed work will depart from previous efforts in that the use of a CAD-centric approach and the development of more sophisticated mesh deformation algorithms will make it possible to include wing planform design variables, which is likely to result in dramatic design changes being applied during optimization. The optimization will be performed with the same SQP algorithm (SNOPT) that was previously mentioned.

Initially, we will optimize aircraft configurations by maximizing range (which can be given as a simple function of drag coefficient and structural weight), subject to stress constraints at a maneuver condition. As previously mentioned, an automated treatment of multipoint problems is needed for practical design, since aircraft, in practice, operate at different altitudes, speeds and load. In the multidisciplinary case, Pareto fronts can also be generated to quantify trade-offs.

Since the aeroservoelastic model consists of a two-dimensional, two degree of freedom problem, a critical wing section will be selected from the wing to solve the flutter suppression problem. The resulting designs from the aero-structural optimization will provide the stiffness characteristics and aerodynamic forces for the aeroservoelasticity model, which will analyze the flutter characteristics and design a control system that maximizes the flutter speed.