Search NASA⌕ Search

SEARCH · Search NASA

Results for “objective function”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Machine Learning Techniques in Optimal Design

Many important applications can be formalized as constrained optimization tasks. For example, we are studying the engineering domain of two-dimensional (2-D) structural design. In this task, the goal is to design a structure of minimum weight that bears a set of loads. A solution to a design problem in which there is a single load (L) and two stationary support points (S1 and S2) consists of four members, E1, E2, E3, and E4 that connect the load to the support points is discussed. In principle, optimal solutions to problems of this kind can be found by numerical optimization techniques. However, in practice [Vanderplaats, 1984] these methods are slow and they can produce different local solutions whose quality (ratio to the global optimum) varies with the choice of starting points. Hence, their applicability to real-world problems is severely restricted. To overcome these limitations, we propose to augment numerical optimization by first performing a symbolic compilation stage to produce: (a) objective functions that are faster to evaluate and that depend less on the choice of the starting point and (b) selection rules that associate problem instances to a set of recommended solutions. These goals are accomplished by successive specializations of the problem class and of the associated objective functions. In the end, this process reduces the problem to a collection of independent functions that are fast to evaluate, that can be differentiated symbolically, and that represent smaller regions of the overall search space. However, the specialization process can produce a large number of sub-problems. This is overcome by deriving inductively selection rules which associate problems to small sets of specialized independent sub-problems. Each set of candidate solutions is chosen to minimize a cost function which expresses the tradeoff between the quality of the solution that can be obtained from the sub-problem and the time it takes to produce it. The overall solution to the problem, is then obtained by solving in parallel each of the sub-problems in the set and computing the one with the minimum cost. In addition to speeding up the optimization process, our use of learning methods also relieves the expert from the burden of identifying rules that exactly pinpoint optimal candidate sub-problems. In real engineering tasks it is usually too costly to the engineers to derive such rules. Therefore, this paper also contributes to a further step towards the solution of the knowledge acquisition bottleneck [Feigenbaum, 1977] which has somewhat impaired the construction of rulebased expert systems.

Cerbone, Giuseppe↗
NASA NTRS1993

Rotor design optimization using a free wake analysis

The aim of this effort was to develop a comprehensive performance optimization capability for tiltrotor and helicopter blades. The analysis incorporates the validated EHPIC (Evaluation of Hover Performance using Influence Coefficients) model of helicopter rotor aerodynamics within a general linear/quadratic programming algorithm that allows optimization using a variety of objective functions involving the performance. The resulting computer code, EHPIC/HERO (HElicopter Rotor Optimization), improves upon several features of the previous EHPIC performance model and allows optimization utilizing a wide spectrum of design variables, including twist, chord, anhedral, and sweep. The new analysis supports optimization of a variety of objective functions, including weighted measures of rotor thrust, power, and propulsive efficiency. The fundamental strength of the approach is that an efficient search for improved versions of the baseline design can be carried out while retaining the demonstrated accuracy inherent in the EHPIC free wake/vortex lattice performance analysis. Sample problems are described that demonstrate the success of this approach for several representative rotor configurations in hover and axial flight. Features that were introduced to convert earlier demonstration versions of this analysis into a generally applicable tool for researchers and designers is also discussed.

Quackenbush, Todd R.↗
NASA NTRS1990

Optimization of actuator and sensor placement for on-orbit identification in large flexible spacecraft

There has been considerable research on choosing actuator and sensor locations in large flexible spacecraft in order to optimize the controllability and observability of the system, or to maximize some objective function of control system performance. Future large flexible spacecraft may require on-orbit identification of the structure to tune the control system, because such tests cannot be performed in a one-g environment before launch. This indicates that the choice of actuator and sensor locations must serve a dual purpose, for control and for identification. This paper develops concepts for a degree of identifiability and studies placement of actuators and sensors on a free-free beam to optimize such objective functions. The results in this simple situation suggest that in free-free spacecraft structures in orbit, placement for control and placement for identification may often be consistent objectives rather than conflicting objectives.

Bergmann, Martin↗
NASA NTRS1980

A hill-sliding strategy for initialization of Gaussian clusters in the multidimensional space

A hill sliding technique was devised to extract Gaussian clusters from the multivariate probability density estimate of sample data for the first step of iterative unsupervised classification. Each cluster was assumed to posses a unimodal normal distribution. A clustering function proposed distinguished elements of a cluster under formation from the rest in the feature space. Initial clusters were extracted one by one according to the hill sliding tactics. A dimensionless cluster compactness parameter was proposed as a universal measure of cluster goodness and used satisfactorily in test runs with LANDSAT multispectral scanner data. The normalized divergence, defined by the cluster divergence divided by the entropy of the entire sample data, was utilized as a general separability measure between clusters. An overall clustering objective function was set forth in terms of cluster covariance matrices, from which the cluster compactness measure could be deduced. Minimal improvement of initial data partitioning was evaluated by this objective function in eliminating scattered sparse data points. The hill sliding clustering technique developed herein has the potential applicability to decomposition any multivariate mixture distribution into a number of unimodal distributions when an appropriate distribution function to the data set is employed.

Park, J. K.↗
NASA NTRS2004

On Improving Efficiency of Differential Evolution for Aerodynamic Shape Optimization Applications

Differential Evolution (DE) is a simple and robust evolutionary strategy that has been provEn effective in determining the global optimum for several difficult optimization problems. Although DE offers several advantages over traditional optimization approaches, its use in applications such as aerodynamic shape optimization where the objective function evaluations are computationally expensive is limited by the large number of function evaluations often required. In this paper various approaches for improving the efficiency of DE are reviewed and discussed. Several approaches that have proven effective for other evolutionary algorithms are modified and implemented in a DE-based aerodynamic shape optimization method that uses a Navier-Stokes solver for the objective function evaluations. Parallelization techniques on distributed computers are used to reduce turnaround times. Results are presented for standard test optimization problems and for the inverse design of a turbine airfoil. The efficiency improvements achieved by the different approaches are evaluated and compared.

Madavan, Nateri K.↗
NASA NTRS1979

Adaptable structural synthesis using advanced analysis and optimization coupled by a computer operating system

A finite element program is linked with a general purpose optimization program in a 'programing system' which includes user supplied codes that contain problem dependent formulations of the design variables, objective function and constraints. The result is a system adaptable to a wide spectrum of structural optimization problems. In a sample of numerical examples, the design variables are the cross-sectional dimensions and the parameters of overall shape geometry, constraints are applied to stresses, displacements, buckling and vibration characteristics, and structural mass is the objective function. Thin-walled, built-up structures and frameworks are included in the sample. Details of the system organization and characteristics of the component programs are given.

Sobieszczanski-Sobieski, J.↗
NASA NTRS1994

Local performance optimization for a class of redundant eight-degree-of-freedom manipulators

Local performance optimization for joint limit avoidance and manipulability maximization (singularity avoidance) is obtained by using the Jacobian matrix pseudoinverse and by projecting the gradient of an objective function into the Jacobian null space. Real-time redundancy optimization control is achieved for an eight-joint redundant manipulator having a three-axis spherical shoulder, a single elbow joint, and a four-axis spherical wrist. Symbolic solutions are used for both full-Jacobian and wrist-partitioned pseudoinverses, partitioned null-space projection matrices, and all objective function gradients. A kinematic limitation of this class of manipulators and the limitation's effect on redundancy resolution are discussed. Results obtained with graphical simulation are presented to demonstrate the effectiveness of local redundant manipulator performance optimization. Actual hardware experiments performed to verify the simulated results are also discussed. A major result is that the partitioned solution is desirable because of low computation requirements. The partitioned solution is suboptimal compared with the full solution because translational and rotational terms are optimized separately; however, the results show that the difference is not significant. Singularity analysis reveals that no algorithmic singularities exist for the partitioned solution. The partitioned and full solutions share the same physical manipulator singular conditions. When compared with the full solution, the partitioned solution is shown to be ill-conditioned in smaller neighborhoods of the shared singularities.

Williams, Robert L., II↗
NASA NTRS1995

Integrated design and manufacturing for the high speed civil transport (a combined aerodynamics/propulsion optimization study)

This report documents the efforts of a Georgia Tech High Speed Civil Transport (HSCT) aerospace student design team in completing a design methodology demonstration under NASA's Advanced Design Program (ADP). Aerodynamic and propulsion analyses are integrated into the synthesis code FLOPS in order to improve its prediction accuracy. Executing the integrated product and process development (IPPD) methodology proposed at the Aerospace Systems Design Laboratory (ASDL), an improved sizing process is described followed by a combined aero-propulsion optimization, where the objective function, average yield per revenue passenger mile ($/RPM), is constrained by flight stability, noise, approach speed, and field length restrictions. Primary goals include successful demonstration of the application of the response surface methodolgy (RSM) to parameter design, introduction to higher fidelity disciplinary analysis than normally feasible at the conceptual and early preliminary level, and investigations of relationships between aerodynamic and propulsion design parameters and their effect on the objective function, $/RPM. A unique approach to aircraft synthesis is developed in which statistical methods, specifically design of experiments and the RSM, are used to more efficiently search the design space for optimum configurations. In particular, two uses of these techniques are demonstrated. First, response model equations are formed which represent complex analysis in the form of a regression polynomial. Next, a second regression equation is constructed, not for modeling purposes, but instead for the purpose of optimization at the system level. Such an optimization problem with the given tools normally would be difficult due to the need for hard connections between the various complex codes involved. The statistical methodology presents an alternative and is demonstrated via an example of aerodynamic modeling and planform optimization for a HSCT.

Baecher, Juergen↗
NASA NTRS2021

Compact Augmented Spark Igniters for Liquid Rocket Engines

The objective of this dissertation is to characterize augmented spark impinging (torch) igniters. Torch igniters inject propellants into a small prechamber where a spark igniter imparts the activation energy to initiate combustion. The flame stabilizes within the prechamber and exits into the main combustion chamber where it ignites the main flow directly or by mixing with a coaxial flow to create a secondary flame. The underlying phenomena that promote reliable ignition are multifaceted and required a broad scope of investigations. Initial efforts focused on spark discharges across an annular geometry. Gaseous propellent projects the electrical arc toward the prechamber, where the arc and the thermal energy is imparted to the surrounding fluid then meets with a combustible mixture. Discharges are parametrically examined against pressure, spark-gap size, and exciter types to emulate transient engine startup. The energy imparted to the flow, approximate velocity, and distance traveled by the exhaust plume are determined. Spark quenching (no energy deposition) and its effects on perceived versus actual measurements are shown. Results outline the prechamber target zone where a combustible mixture is necessary for ignition. Next, three-dimensional, time-accurate, and nonreacting computational fluid dynamics simulations assess effects of geometric and mass flow differences on the size, location, and composition of a combustible mixture. Fluid dynamic phenomena are elucidated. Correlations between nondimensional variables are found. Results are coupled with a conservation-law analysis to produce novel objective functions. Lastly, a full-scale, modular torch igniter is tested. Injector sizes, mixture ratios, and injector momentum ratios are systematically varied to map ignition probability and ignition delay. Results show the importance of local mixture ratios, provide injector momentum ratios guidelines, and demonstrate reliable ignition of core mixture ratios typically outside flammability limits. The deliverable of this project is not a torch igniter or derivative thereof. Instead, this dissertation produces novel objective-functions for igniter design. These equations are functions of variables that may be constrained by engine-level requirements, physics, and experimental results from this work. They provide a foundation for future igniter design and will minimize iterative processes from the design cycle for next-generation ignition hardware.

Torch Igniter↗
NASA NTRS1983

Aeroelastic tailoring of rotor blades for vibration reduction in forward flight

Modern structural optimization techniques are applied to vibration reduction of helicopter rotor blades in forward flight. The objective function minimized consists of the oscillatory vertical hub shears or the hub rolling moments at one particular advance ratio. The behavior constraints are the frequency placements of the blade and the requirement that aeroelastic stability margins, in hover, remain unaffected by the optimization process. The aeroelastic stability and response analysis is based on a fully coupled flap-lag-torsional analysis of the blade. The vertical hub shears and rolling moments used as the objective function are obtained by appropriate integration of the loads acting along the span of the blade combined with a transformation to a hub fixed coordinate system, and a summation over the total number of blades. Numerical results for both a stiff-in-plane and a soft-in-plane configuration are presented, indicating that structural optimization yields the highest benefits when applied to soft-in-plane blade configuration. The results indicate substantial (15-40 percent) reduction in vibration levels, as well as a blade which is 20 percent lighter than the initial design.

Friedmann, P. P.↗
NASA NTRS1999

Demonstration of Automatically-Generated Adjoint Code for Use in Aerodynamic Shape Optimization

Gradient-based optimization requires accurate derivatives of the objective function and constraints. These gradients may have previously been obtained by manual differentiation of analysis codes, symbolic manipulators, finite-difference approximations, or existing automatic differentiation (AD) tools such as ADIFOR (Automatic Differentiation in FORTRAN). Each of these methods has certain deficiencies, particularly when applied to complex, coupled analyses with many design variables. Recently, a new AD tool called ADJIFOR (Automatic Adjoint Generation in FORTRAN), based upon ADIFOR, was developed and demonstrated. Whereas ADIFOR implements forward-mode (direct) differentiation throughout an analysis program to obtain exact derivatives via the chain rule of calculus, ADJIFOR implements the reverse-mode counterpart of the chain rule to obtain exact adjoint form derivatives from FORTRAN code. Automatically-generated adjoint versions of the widely-used CFL3D computational fluid dynamics (CFD) code and an algebraic wing grid generation code were obtained with just a few hours processing time using the ADJIFOR tool. The codes were verified for accuracy and were shown to compute the exact gradient of the wing lift-to-drag ratio, with respect to any number of shape parameters, in about the time required for 7 to 20 function evaluations. The codes have now been executed on various computers with typical memory and disk space for problems with up to 129 x 65 x 33 grid points, and for hundreds to thousands of independent variables. These adjoint codes are now used in a gradient-based aerodynamic shape optimization problem for a swept, tapered wing. For each design iteration, the optimization package constructs an approximate, linear optimization problem, based upon the current objective function, constraints, and gradient values. The optimizer subroutines are called within a design loop employing the approximate linear problem until an optimum shape is found, the design loop limit is reached, or no further design improvement is possible due to active design variable bounds and/or constraints. The resulting shape parameters are then used by the grid generation code to define a new wing surface and computational grid. The lift-to-drag ratio and its gradient are computed for the new design by the automatically-generated adjoint codes. Several optimization iterations may be required to find an optimum wing shape. Results from two sample cases will be discussed. The reader should note that this work primarily represents a demonstration of use of automatically- generated adjoint code within an aerodynamic shape optimization. As such, little significance is placed upon the actual optimization results, relative to the method for obtaining the results.

Green, Lawrence↗
NASA NTRSDate not supplied

End-To-End Uncertainty Quantification with Analytical Derivatives for Design Under Uncertainty

Uncertainty quantification (UQ) is a rapidly growing and evolving discipline, especially within the aerospace community. Performing analysis with UQ can provide decision makers with a wealth of information about a candidate design. However, the value of UQ is fully realized when the information gained during UQ analysis is leveraged in a feedback loop of a design optimization process, often referred to as design under uncertainty. Although design under uncertainty can be a powerful risk mitigation technique, there are a number of roadblocks that prevent its implementation. Two primary factors are computational costs and added complexity of the analysis. High fidelity simulations on the order tens of uncertain variables quickly become computationally infeasible. Also, implementing UQ into an existing multidisciplinary design and optimization (MDO) process often requires extensive knowledge of the UQ methods and careful treatment of the problem formulation. The objective of this work is to address these two primary roadblocks and enable practitioners to efficiently perform design under uncertainty with limited knowledge of the UQ discipline. Methods outlined in this paper demonstrate MDO incorporating UQ into the design process, leveraging an analytic derivative tool chain through the entire optimization. The proposed approach leverages machine learning techniques to generate a differentiable confidence interval output from polynomial chaos models. This technique, coupled with the incorporation of analytical derivatives through the Polynomial Chaos Expansion (PCE) process, eliminates the need to estimate derivatives which are usually obtained from finite difference, complex step, or similar methods. Developing a differentiable confidence interval allows mixed uncertainty problems (both epistemic and aleatory) to be modeled. Without such modeling, these problems cannot accurately predict objective functions containing statistical quantities such as mean and variance. The addition of analytic derivatives to a polynomial chaos-based UQ method decreases the computational costs of performing design under uncertainty by orders of magnitude in comparison with methods such as complex step. The method and codes developed are modular in nature and are a drop-in solution for design under uncertainty within existing MDO problems. A low-fidelity analytical multidisciplinary optimization under uncertainty for a wing design in OpenMDAO is detailed in this paper. This demonstration case will include both objective functions and constraints which are influenced by uncertain parameters.

Ben D Phillips↗
NASA NTRSDate not supplied

End-To-End Uncertainty Quantification with Analytical Derivatives for Design Under Uncertainty

Uncertainty quantification (UQ) is a rapidly growing and evolving discipline, especially within the aerospace community. Performing analysis with UQ can provide decision makers with a wealth of information about a candidate design. However, the value of UQ is fully realized when the information gained during UQ analysis is leveraged in a feedback loop of a design optimization process, often referred to as design under uncertainty. Although design under uncertainty can be a powerful risk mitigation technique, there are a number of roadblocks that prevent its implementation. Two primary factors are computational costs and added complexity of the analysis. High fidelity simulations on the order tens of uncertain variables quickly become computationally infeasible. Also, implementing UQ into an existing multidisciplinary design and optimization (MDO) process often requires extensive knowledge of the UQ methods and careful treatment of the problem formulation. The objective of this work is to address these two primary roadblocks and enable practitioners to efficiently perform design under uncertainty with limited knowledge of the UQ discipline. Methods outlined in this paper demonstrate MDO incorporating UQ into the design process, leveraging an analytic derivative tool chain through the entire optimization. The proposed approach leverages machine learning techniques to generate a differentiable confidence interval output from polynomial chaos models. This technique, coupled with the incorporation of analytical derivatives through the Polynomial Chaos Expansion (PCE) process, eliminates the need to estimate derivatives which are usually obtained from finite difference, complex step, or similar methods. Developing a differentiable confidence interval allows mixed uncertainty problems (both epistemic and aleatory) to be modeled. Without such modeling, these problems cannot accurately predict objective functions containing statistical quantities such as mean and variance. The addition of analytic derivatives to a polynomial chaos-based UQ method decreases the computational costs of performing design under uncertainty by orders of magnitude in comparison with methods such as complex step. The method and codes developed are modular in nature and are a drop-in solution for design under uncertainty within existing MDO problems. A low-fidelity analytical multidisciplinary optimization under uncertainty for a wing design in OpenMDAO is detailed in this paper. This demonstration case will include both objective functions and constraints which are influenced by uncertain parameters.

Ben Phillips↗
NASA NTRS1989

Efficient sensitivity analysis and optimization of a helicopter rotor

Aeroelastic optimization of a system essentially consists of the determination of the optimum values of design variables which minimize the objective function and satisfy certain aeroelastic and geometric constraints. The process of aeroelastic optimization analysis is illustrated. To carry out aeroelastic optimization effectively, one needs a reliable analysis procedure to determine steady response and stability of a rotor system in forward flight. The rotor dynamic analysis used in the present study developed inhouse at the University of Maryland is based on finite elements in space and time. The analysis consists of two major phases: vehicle trim and rotor steady response (coupled trim analysis), and aeroelastic stability of the blade. For a reduction of helicopter vibration, the optimization process requires the sensitivity derivatives of the objective function and aeroelastic stability constraints. For this, the derivatives of steady response, hub loads and blade stability roots are calculated using a direct analytical approach. An automated optimization procedure is developed by coupling the rotor dynamic analysis, design sensitivity analysis and constrained optimization code CONMIN.

Lim, Joon W.↗
NASA NTRS1992

Structural Tailoring of Advanced Turboprops (STAT). Theoretical manual

This manual describes the theories in the Structural Tailoring of Advanced Turboprops (STAT) computer program, which was developed to perform numerical optimizations on highly swept propfan blades. The optimization procedure seeks to minimize an objective function, defined as either direct operating cost or aeroelastic differences between a blade and its scaled model, by tuning internal and external geometry variables that must satisfy realistic blade design constraints. The STAT analyses include an aerodynamic efficiency evaluation, a finite element stress and vibration analysis, an acoustic analysis, a flutter analysis, and a once-per-revolution (1-p) forced response life prediction capability. The STAT constraints include blade stresses, blade resonances, flutter, tip displacements, and a 1-P forced response life fraction. The STAT variables include all blade internal and external geometry parameters needed to define a composite material blade. The STAT objective function is dependent upon a blade baseline definition which the user supplies to describe a current blade design for cost optimization or for the tailoring of an aeroelastic scale model.

Brown, K. W.↗
NASA NTRS1984

A new implementation of the programming system for structural synthesis (PROSSS-2)

This new implementation of the PROgramming System for Structural Synthesis (PROSSS-2) combines a general-purpose finite element computer program for structural analysis, a state-of-the-art optimization program, and several user-supplied, problem-dependent computer programs. The results are flexibility of the optimization procedure, organization, and versatility of the formulation of constraints and design variables. The analysis-optimization process results in a minimized objective function, typically the mass. The analysis and optimization programs are executed repeatedly by looping through the system until the process is stopped by a user-defined termination criterion. However, some of the analysis, such as model definition, need only be one time and the results are saved for future use. The user must write some small, simple FORTRAN programs to interface between the analysis and optimization programs. One of these programs, the front processor, converts the design variables output from the optimizer into the suitable format for input into the analyzer. Another, the end processor, retrieves the behavior variables and, optionally, their gradients from the analysis program and evaluates the objective function and constraints and optionally their gradients. These quantities are output in a format suitable for input into the optimizer. These user-supplied programs are problem-dependent because they depend primarily upon which finite elements are being used in the model. PROSSS-2 differs from the original PROSSS in that the optimizer and front and end processors have been integrated into the finite element computer program. This was done to reduce the complexity and increase portability of the system, and to take advantage of the data handling features found in the finite element program.

Rogers, James L., Jr.↗
NASA NTRS1992

The role of under-determined approximations in engineering and science application

There is currently a great deal of interest in using response surfaces in the optimization of aircraft performance. The objective function and/or constraint equations involved in these optimization problems may come from numerous disciplines such as structures, aerodynamics, environmental engineering, etc. In each of these disciplines, the mathematical complexity of the governing equations usually dictates that numerical results be obtained from large computer programs such as a finite element method program. Thus, when performing optimization studies, response surfaces are a convenient way of transferring information from the various disciplines to the optimization algorithm as opposed to bringing all the sundry computer programs together in a massive computer code. Response surfaces offer another advantage in the optimization of aircraft structures. A characteristic of these types of optimization problems is that evaluation of the objective function and response equations (referred to as a functional evaluation) can be very expensive in a computational sense. Because of the computational expense in obtaining functional evaluations, the present study was undertaken to investigate under-determinined approximations. An under-determined approximation is one in which there are fewer training pairs (pieces of information about a function) than there are undetermined parameters (coefficients or weights) associated with the approximation. Both polynomial approximations and neural net approximations were examined. Three main example problems were investigated: (1) a function of one design variable was considered; (2) a function of two design variables was considered; and (3) a 35 bar truss with 4 design variables was considered.

Carpenter, William C.↗
NASA NTRS2006

Reentry-Vehicle Shape Optimization Using a Cartesian Adjoint Method and CAD Geometry

A DJOINT solutions of the governing flow equations are becoming increasingly important for the development of efficient analysis and optimization algorithms. A well-known use of the adjoint method is gradient-based shape. Given an objective function that defines some measure of performance, such as the lift and drag functionals, its gradient is computed at a cost that is essentially independent of the number of design variables (e.g., geometric parameters that control the shape). Classic aerodynamic applications of gradient-based optimization include the design of cruise configurations for transonic and supersonic flow, as well as the design of high-lift systems. are perhaps the most promising approach for addressing the issues of flow solution automation for aerodynamic design problems. In these methods, the discretization of the wetted surface is decoupled from that of the volume mesh. This not only enables fast and robust mesh generation for geometry of arbitrary complexity, but also facilitates access to geometry modeling and manipulation using parametric computer-aided design (CAD). In previous work on Cartesian adjoint solvers, Melvin et al. developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the two-dimensional Euler equations using a ghost-cell method to enforce the wall boundary conditions. In Refs. 18 and 19, we presented an accurate and efficient algorithm for the solution of the adjoint Euler equations discretized on Cartesian meshes with embedded, cut-cell boundaries. Novel aspects of the algorithm were the computation of surface shape sensitivities for triangulations based on parametric-CAD models and the linearization of the coupling between the surface triangulation and the cut-cells. The accuracy of the gradient computation was verified using several three-dimensional test cases, which included design variables such as the free stream parameters and the planform shape of an isolated wing. The objective of the present work is to extend our adjoint formulation to problems involving general shape changes. Factors under consideration include the computation of mesh sensitivities that provide a reliable approximation of the objective function gradient, as well as the computation of surface shape sensitivities based on a direct-CAD interface. We present detailed gradient verification studies and then focus on a shape optimization problem for an Apollo-like reentry vehicle. The goal of the optimization is to enhance the lift-to-drag ratio of the capsule by modifying the shape of its heat-shield in conjunction with a center-of-gravity (c.g.) offset. This multipoint and multi-objective optimization problem is used to demonstrate the overall effectiveness of the Cartesian adjoint method for addressing the issues of complex aerodynamic design.

Nemec, Marian↗