David W. Zingg

Multipoint and Multi-Objective Aerodynamic Shape Optimization

A Newton‐Krylov algorithm is presented for the aerodynamic optimization of singleand multi-element airfoil configurations. The flow is governed by the compressible Navier‐Stokes equations in conjunction with a one-equation turbulence model. The preconditioned generalized minimum residual method is applied to solve the discreteadjoint equation, leading to a fast computation of accurate objective function gradients. Optimization constraints are enforced through a penalty formulation, and the resulting unconstrained problem is solved via a quasi-Newton method. Design examples include lift-enhancement and multi-point lift-constrained drag minimization problems. Furthermore, the new algorithm is used to compute a Pareto front for a multi-objective problem, and the results are validated using a genetic algorithm. Overall, the new algorithm provides an ecient and robust approach for addressing the issues of complex aerodynamic

Newton-Krylov Algorithm for Aerodynamic Design Using the Navier-Stokes Equations

A Newton‐Krylov algorithm is presented for two-dimensional Navier‐Stokes aerodynamic shape optimization problems. The algorithm is applied to both the discrete-adjoint and the discrete e ow-sensitivity methods for calculating the gradient of the objective function. The adjoint and e ow-sensitivity equations are solved using a novel preconditioned generalized minimum residual (GMRES)strategy. Together with a complete linearization of the discretized Navier‐Stokes and turbulence model equations, this results in an accurate and efecient evaluation of the gradient. Furthermore, fast e ow solutions are obtained using the same preconditioned GMRES strategy in conjunction with an inexact Newton approach. The performance of the new algorithm is demonstrated for several design examples,includinginversedesign,lift-constraineddragminimization, liftenhancement, and maximization of lift-to-dragratio. In all examples, the normof the gradientisreduced by several ordersof magnitude, indicating that alocalminimumhasbeen obtained. Bytheuseoftheadjoint method,thegradient isobtained infromone-e fth to one-half of the time required to converge a eow solution.

Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous-Approximation Terms

0mesh continuity at block interfaces, accommodates arbitrary block topologies, and has low interblock-communication overhead. The resulting discrete equations are solved iteratively using an inexact-Newton method. At each Newton iteration, the linear system is solved inexactly using a Krylov-subspace iterative method, and both additive Schwarz and approximate Schur preconditioners are investigated. The algorithm is tested on the ONERA M6 wing geometry. We conclude that the approximate Schur preconditioner is an efficient alternative to the Schwarz preconditioner. Overall, the results demonstrate that the Newton–Krylov algorithm is very efficient: using 24 processors, a transonic flow on a 96-block, 1-million-node mesh requires 12 minutes for a 10-order reduction of the residual norm.

Comparison of High-Accuracy Finite-Difference Methods for Linear Wave Propagation

This paper analyzes a number of high-order and optimized finite-difference methods for numerically simulating the propagation and scattering of linear waves, such as electromagnetic, acoustic, and elastic waves. The spatial operators analyzed include compact schemes, noncompact schemes, schemes on staggered grids, and schemes which are optimized to produce specific characteristics. The time-marching methods include Runge--Kutta methods, Adams--Bashforth methods, and the leapfrog method. In addition, the following fully-discrete finite-difference methods are studied: a one-step implicit scheme with a three-point spatial stencil, a one-step explicit scheme with a five-point spatial stencil, and a two-step explicit scheme with a five-point spatial stencil. For each method, the number of grid points per wavelength required for accurate simulation of wave propagation over large distances is presented. The results provide a clear understanding of the relative merits of the methods compared, especially the trade-offs associated with the use of optimized methods. A numerical example is given which shows that the benefits of an optimized scheme can be small if the waveform has broad spectral content.

High-Accuracy Finite-Difference Schemes for Linear Wave Propagation

Two high-accuracy finite-difference schemes for simulating long-range linear wave propagation are presented: a maximum-order scheme and an optimized scheme. The schemes combine a seven-point spatial operator and an explicit six-stage low-storage time-march method of Runge--Kutta type. The maximum-order scheme can accurately simulate the propagation of waves over distances greater than five hundred wavelengths with a grid resolution of less than twenty points per wavelength. The optimized scheme is found by minimizing the maximum phase and amplitude errors for waves which are resolved with at least ten points per wavelength, based on Fourier error analysis. It is intended for simulations in which waves travel under three hundred wavelengths. For such cases, good accuracy is obtained with roughly ten points per wavelength.

A comparative evaluation of genetic and gradient-based algorithms applied to aerodynamic optimization

A genetic algorithm is compared with a gradient-based (adjoint) algorithm in the context of several aerodynamic shape optimization problems. The examples include singlepoint and multipoint optimization problems, as well as the computation of a Pareto front. The results demonstrate that both algorithms converge reliably to the same optimum. Depending on the nature of the problem, the number of design variables, and the degree of convergence, the genetic algorithm requires from 5 to 200 times as many function evaluations as the gradientbased algorithm.

Induced-Drag Minimization of Nonplanar Geometries Based on the Euler Equations

The induced drag of several nonplanar configurations is minimized using an aerodynamic shape optimization algorithm based on the Euler equations. The algorithm is first validated using twist optimization to recover an elliptical lift distribution. Planform optimization reveals that an elliptical planform is not optimal when side-edge separation is present. Optimized winglet and box-wing geometries are found to have span efficiencies that agree well with lifting-line analysis, provided the bound constraints on the entire geometry are accounted for in the linear analyses. For the same spanwise and vertical bound constraints, a nonplanar split-tip geometry outperforms both the winglet and box-wing geometries, because it can more easily maximize the vertical extent at the tip. The performance of all the optimized geometries is verified using refined grids consisting of 88-152 million nodes.

Multimodality and Global Optimization in Aerodynamic Design

Two optimization algorithms are presented that are capable of finding a global optimum in a computationally efficient manner: a gradient-based multistart algorithm based on Sobol sampling and a hybrid optimizer combining a genetic algorithm with a gradient-based algorithm. The optimizers are used to investigate multimodality in aerodynamic-shape-optimization problems. The performance of each algorithm is tested on an analytical test function as well as several aerodynamic-shape-optimization problems in two and three dimensions. In each problem the primary objectives are to classify the problem according to the degree of multimodality and to identify the preferred optimization algorithm for the problem. The results show that multimodality should not always be assumed in aerodynamic-shape-optimization problems. Typical two-dimensional airfoil-optimization problems are unimodal. Three-dimensional shape-optimization problems may contain local optima. The number of local optima tends to increase with increasin...

A generalized framework for nodal first derivative summation-by-parts operators

A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are (i) non-repeating interior point operators, (ii) nonuniform nodal distribution in the computational domain, (iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the SBP property. It is proven that the positive-definite norm matrix of each SBP operator must be associated with a quadrature rule; moreover, given a quadrature rule there exists a corresponding SBP operator, where for diagonal-norm SBP operators the weights of the quadrature rule must be positive. The generalized framework gives a straightforward means of posing many known approximations to the first derivative as SBP operators; several are surveyed, such as discontinuous Galerkin discretizations based on the Legendre-Gauss quadrature points, and shown to be SBP operators. Moreover, the new framework provides a method for constructing SBP operators by starting from quadrature rules; this is illustrated by constructing novel SBP operators from known quadrature rules. To demonstrate the utility of the generalization, the Legendre-Gauss and Legendre-Gauss-Radau quadrature points are used to construct SBP operators that do not include one or both boundary nodes.

Summation-by-parts operators and high-order quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. For diagonal weight matrices, the accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; in particular, the diagonal norm accurately approximates the L^2 norm for functions, and multi-dimensional SBP discretizations accurately approximate the divergence theorem.