Shawn E. Gano

Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function

Solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints. Variable fidelity methods address this issue by using lower-fidelity models and a scaling function to approximate the higher-fidelity models in a provably convergent framework. In the past, scaling functions have mainly been either first-order multiplicative or additive corrections. These are being extended to second order. In this investigation variable metric approaches for calculating second-order scaling information are developed. A kriging-based scaling function is introduced to better approximate the high-fidelity response on a more global level. An adaptive hybrid method is also developed in this investigation. The adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization. The methodologies developed in this research are compared to existing methods using two demonstration problems. The first problem is analytic, whereas the second involves the design of a supercritical high-lift airfoil. The results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high-fidelity function calls required for convergence. The results also indicate the hybrid method is both robust and effective.

Variable Fidelity Optimization Using a Kriging Based Scaling Function

Solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods may not be feasible given design cycle time constraints. Variable fldelity methods address this issue by using lower fldelity models and a scaling function to approximate the higher fldelity models in a provably convergent framework. In the past, scaling functions have mainly been either flrst order multiplicative or additive corrections; recently these have been extended to second order. In this work a kriging based scaling function is introduced to better approximate the high fldelity response on a more global level. An adaptive hybrid method is also studied. It combines the additive and multiplicative approaches so that the designer doesn’t have to determine which is better before optimization. The difierent methods are theoretically described and then compared using two demonstration problems. The flrst problem is analytic, while the other is a design of a supercritical high-lift airfoil. The results show that the warm-started kriging based scaling methods have the potential to improve computational expense by lowering the number of high fldelity function calls required for convergence. The results also indicate the hybrid method is efiective.

OPTIMIZED UNMANNED AERIAL VEHICLE WITH WING MORPHING FOR EXTENDED RANGE AND ENDURANCE

Due to their current successes, unmanned aerial vehicles (UAVs) are becoming a standard means of collecting information. However, as their missions become more complex and require them to fly farther, UAVs can become large and expensive due to fuel needs. Sidestepping the paradigm of a fixed static wing, the variform concept developed in this paper allows for greater fuel efficiency. Bulky wings could morph into sleeker profiles, reducing drag, as they burn fuel. The development of such wings will rely heavily on computational design exploiting state of the art optimization techniques that account for uncertainty and insure reliability.

Morphing UAV Pareto Curve Shift for Enhanced Performance

Research in unmanned aerial vehicles (UAVs) has grown in interest over the past couple decades. Historically, UAVs were designed to maximize endurance and range, but demands for UAV designs have changed in recent years. In addition to the traditional demands for endurance and range, today customer demands include maneuverability. Therefore, UAVs are being designed to morph, to change their geometrical shape during flight, for enhanced maneuvering capability. In this investigation the morphing UAV concept under study is referred to as the buckle wing. The design of the buckle-wing airfoil geometries is posed as a multilevel, multiobjective optimization problem. This buckle-wing design problem includes two competing objectives of maneuverability and long range/endurance. Multiobjective problems have many optimal solutions each depicting a dierent compromise scenario. Each optimal solution is a Pareto point, and the set of all these points represents the Pareto curve. This is a powerful means of showing the global picture of the solution eld. The goal of this paper is to explore and compare the Pareto curves of the buckle-wing UAV to that of a conventional non-morphing UAV. In order to make this performance comparison, Compromise Programming is used as the optimizing method, and the VortexPanel Method is used in calculating the aerodynamics. The buckle-wing UAV’s enhanced capabilities are demonstrated both quantitatively and graphically.

Shape Optimization for Conforming Airfoils

Interest in the design and development of unmanned aerial vehicles (UAVs) has increased dramatically in the last two and a half decades. The Buckle-Wing UAV concept being developed in this research is designed to “morph” in a way which facilitates variations in wing loading, aspect ratio and wing section shapes. The Buckle-Wing consists of two highly elastic beam-like lifting surfaces joined at the outboard wing tips in either a pinned or clamped configuration. The Buckle-Wing UAV is capable of morphing between a separated wing configuration designed for maneuverability to a single fixed wing configuration designed for long range/high endurance. The design of the Buckle-Wing’s aerodynamic shapes is critical to the functioning of this adaptive UAV airframe. The airfoils must be capable of functioning both as independent lifting surfaces and as a fused single wing. The adaptive airframe of the BuckleWing requires that the two airfoils/wings conform as one single wing for the extended range and/or endurance configuration. This paper is focused on the use of shape optimization technologies to optimally tailor the aerodynamic performance of the UAV airfoils in both the separated and single fixed wing configuration. A conforming multi-objective and multilevel airfoil shape optimization problem is formulated and solved. Given an exterior airfoil, optimized for long endurance, shape optimization can be used to decomposed the exterior airfoil into two conforming airfoils in such a way that when separated the airfoils produce a 85% increase in lift providing improved maneuverability. ∗Graduate Research Assistant, Student Member AIAA †Professor, Associate Fellow AIAA. Nomenclature α Angle of attack ρ∞ Free stream density ω Turning rate c Thrust-specific fuel consumption cd Drag coefficient cl Lift coefficient D Total drag E Endurance g Acceleration of gravity L Total lift n Wing load factor r Turning radius R Range S Planform Area t Time V∞ Free stream velocity W Weight of aircraft at any given time W1 Weight of aircraft without fuel and with full payload Wo Weight of aircraft with full fuel and payload

Reliability-based design using variable-fidelity optimization

Competitive marketplaces have driven the need for simulation-based design optimization to produce efficient and cost-effective designs. However, such design practices typically do not take into account model uncertainties or manufacturing tolerances. Such designs may lie on failure-driven constraints and are characterized by a high probability of failure. Reliability-based design optimization (RBDO) methods have been developed to obtain designs that optimize a merit function while ensuring a target reliability level is satisfied. Unfortunately, these methods are notorious for the high computational expense they require to converge. In this research variable-fidelity methods are used to reduce the cost of RBDO. Variable-fidelity methods use a set of models with varying degrees of fidelity and computational expense to aid in reducing the cost of optimization. The variable-fidelity RBDO methodology developed in this investigation is demonstrated on two test cases: a nonlinear analytic problem and a high-lift airfoil design problem. For each of these problems the proposed method shows considerable savings for performing RBDO as compared with standard approaches.

Optimum design of an interbody implant for lumbar spine fixation

A new minimally invasive surgical technique for lumbar spine fixation is currently in development. The procedure makes use of an interbody implant that is inserted between two vertebral bodies. The implant is packed with bone graft material that fuses the motion segment. The implant must be capable of retaining bone graft material and supporting spinal loads while fusion occurs. The different load conditions analyzed include: compression, flexion, extension, and lateral bending. The goal of this research is to obtain an optimum design of this interbody implant. Finite element-based optimization techniques are used to drive the design. The multiobjective optimization process is performed in two stages: topology optimization followed by shape optimization. As a result, the final design maximizes the volume allocated for the bone graft material and maintains von Mises stress levels in the implant below the stress limit. The finite element-based optimization software GENESIS is used in the design process.

Homotopy methods for constraint relaxation in unilevel reliability based design optimization

In reliability based design optimization, a methodology for finding optimized designs characterized with a low probability of failure the main objective is to minimize a merit function while satisfying the reliability constraints. Traditionally, these have been formulated as a double-loop (nested) optimization problem, which is computationally intensive. A new efficient unilevel formulation for reliability based design optimization was developed by the authors in earlier studies, where the lower-level optimization was replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper-level optimization and imposed as equality constraints. But as most commercial optimizers are usually numerically unreliable when applied to problems accompanied by many equality constraints, an optimization framework for reliability based design using the unilevel formulation is developed here. Homotopy methods are used for constraint relaxation and to obtain a relaxed feasible design and heuristic scheme is employed to update the homotopy parameter.

Update Strategies for Kriging Models for Use in Variable Fidelity Optimization

Many optimization methods for simulation-based design rely on the sequential use of metamodels to reduce the associated computational burden. In particular, kriging models are frequently used in variable fidelity optimization. Nevertheless, such methods may become computationally inefficient when solving problems with large numbers of design variables and/or sampled data points due to the expensive process of optimizing the kriging model parameters each iteration. One solution to this problem would be to replace the kriging models with traditional Taylor series response surface models. Kriging models, however, have been shown to provide good approximations of computer simulations that incorporate larger amounts of data, resulting in better global accuracy. In this paper two metamodel update management schemes (MUMS) are proposed to reduce the cost of using kriging models sequentially by updating the kriging model parameters only when they produce a poor approximation. The two schemes differ in how they determine when the parameters should be updated. The first method uses ratios of likelihood values (LMUMS), which are computed based on the model parameters and the data points used to construct the kriging model. The second scheme uses the trust region ratio (TR-MUMS), which is a ratio that compares the approximation to the true model. Two demonstration problems are used to evaluate the proposed methods: an internal combustion engine sizing problem and a control-augmented structural design problem. The results indicate that the L-MUMS approach does not perform well. The TR-MUMS approach, however, was found to be very effective; on the demonstration problems, it reduced the number of likelihood evaluations by three orders of magnitude compared to using a global optimizer to find the kriging parameters every iteration. It was also found that in trust region-based methods, the kriging model parameters need not be updated using a global optimizer–local methods perform just as well in terms of providing a good approximation without effecting the overall convergence rate, which, in turn, results in a faster execution time.

DEVELOPMENT AND VERIFICATION OF A MATLAB DRIVER FOR THE SNOPT OPTIMIZATION SOFTWARE

The MATLAB program and computing language has seen increased usage both in industry and academia in recent years. This is due to the ease in which it handles matrices and numerical computations. This computing environment also has an array of toolboxes for different mathematical and engineering tasks (e.g., controls, optimization). These toolboxes provide a general suite of numerical tools within a specific discipline for the user. The toolbox codes are general tools and are not typically as robust or as efficient as state of the art numerical codes develop by advanced users in a given discipline. In this research a MATLAB driver which links an existing robust and efficient optimization program SNOPT is developed and tested. The resulting program and driver have proved to be more efficient than the existing MATLAB toolbox codes for optimization.