Joaquim R. R. A. Martins

Multidisciplinary Design Optimization: A Survey of Architectures

Multidisciplinary design optimization is a field of research that studies the application of numerical optimization techniques to the design of engineering systems involving multiple disciplines or components. Since the inception of multidisciplinary design optimization, various methods (architectures) have been developed and applied to solve multidisciplinary design-optimization problems. This paper provides a survey of all the architectures that have been presented in the literature so far. All architectures are explained in detail using a unified description that includes optimization problem statements, diagrams, and detailed algorithms. The diagrams show both data and process flow through the multidisciplinary system and computational elements, which facilitate the understanding of the various architectures, and how they relate to each other. A classification of the multidisciplinary design-optimization architectures based on their problem formulations and decomposition strategies is also provided, a...

The complex-step derivative approximation

The complex-step derivative approximation and its application to numerical algorithms are presented. Improvements to the basic method are suggested that further increase its accuracy and robustness and unveil the connection to algorithmic differentiation theory. A general procedure for the implementation of the complex-step method is described in detail and a script is developed that automates its implementation. Automatic implementations of the complex-step method for Fortran and C/C++ are presented and compared to existing algorithmic differentiation tools. The complex-step method is tested in two large multidisciplinary solvers and the resulting sensitivities are compared to results given by finite differences. The resulting sensitivities are shown to be as accurate as the analyses. Accuracy, robustness, ease of implementation and maintainability make these complex-step derivative approximation tools very attractive options for sensitivity analysis.

High-Fidelity Aerostructural Design Optimization of a Supersonic Business Jet

This paper focuses on the demonstration of an integrated aerostructural method for the design of aerospace vehicles. Both aerodynamics and structures are represented using high-fidelity models such as the Euler equations for the aerodynamics and a detailed finite element model for the primary structure. The aerodynamic outer-mold line and a structure of fixed topology are parameterized using a large number of design variables. The aerostructural sensitivities of aerodynamic and structural cost functions with respect to both outer-mold line shape and structural variables are computed using an accurate and efficient coupled-adjoint procedure. Kreisselmeier‐ Steinhauser functions are used to reduce the number of structural constraints in the problem. Results of the aerodynamic shape and structural optimization of a natural laminar-flow supersonic business jet are presented together with an assessment of the accuracy of the sensitivity information obtained using the coupled-adjoint procedure.

A Coupled-Adjoint Sensitivity Analysis Method for High-Fidelity Aero-Structural Design

This paper presents an adjoint method for sensitivity analysis that is used in an aero-structural aircraft design framework. The aero-structural analysis uses high-fidelity models of both the aerodynamics and the structures. Aero-structural sensitivities are computed using a coupled-adjoint approach that is based on previously developed single discipline sensitivity analysis. Alternative strategies for coupled sensitivity analysis are also discussed. The aircraft geometry and a structure of fixed topology are parameterized using a large number of design variables. The aero-structural sensitivities of aerodynamic and structural functions with respect to these design variables are computed and compared with results given by the complex-step derivative approximation. The coupled-adjoint procedure is shown to yield very accurate sensitivities and to be computationally efficient, making high-fidelity aero-structural design feasible for problems with thousands of design variables.

Scalable Parallel Approach for High-Fidelity Steady-State Aeroelastic Analysis and Adjoint Derivative Computations

Aeroelastic systems achieve the best performance when the aerodynamic shape and structural sizing are optimized concurrently, but such an optimization is challenging when high-fidelity aerodynamic and structural models are required. This paper addresses this challenge through several significant improvements. Fully coupled Newton–Krylov methods are presented for the solution of aerostructural systems and for the corresponding adjoint systems. The coupled adjoint method presented can compute gradients with respect to thousands of multidisciplinary design variables accurately and efficiently. This is enabled by several improvements in the computation of the multidisciplinary terms in the coupled adjoint. The parallel scalability of the methods is demonstrated for a full aircraft configuration using an Euler computational fluid dynamics model with more than 8×106 state variables and a detailed structural finite element model of the wing with more than 1×106 degrees of freedom. The coupled Newton–Krylov metho...

AN AUTOMATED METHOD FOR SENSITIVITY ANALYSIS USING COMPLEX VARIABLES

The complex-step method for calculating sensitivities and its use in numerical algorithms is presented. A general procedure for the implementation of this method is described in detail and a script is developed that automates its implementation. The numerical examples include the automatic conversion of a structural finite element and a two-dimensional computational fluid dynamics code. In both of these examples, the complex-step method is compared with other existing methods, namely finitedierencing, automatic dierentiation and an analytic method. The complex-step method is shown to have implementation advantages over automatic dierentiation and computational advantages over finite-dierencin g.

ADjoint : An Approach for the Rapid Development of Discrete Adjoint Solvers

An automatic differentiation tool is used to develop the adjoint code for a three-dimensional computational fluid dynamics solver. Rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver, we have applied it selectively to produce code that computes the flux Jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method. The resulting linear discrete adjoint system is then solved using the portable, extensible toolkit for scientific computation. This selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint (ADjoint) approach. This approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions, and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver. Furthermore, the approach is largely automatic, thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations. These significant advantages come at the cost of increased memory requirements for the adjoint solver. Derivatives of drag and lift coefficients are validated, and the low computational cost and ease of implementation of the method are shown.

Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models

This paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework. This framework hinges on a new equation, the unifying chain rule, from which all the methods can be derived. The computation of derivatives is described as a two-step process: the evaluation of the partial derivatives and the computation of the total derivatives, which are dependent on the partial derivatives. Finite differences, the complex-step method, and symbolic differentiation are discussed as options for computing the partial derivatives. It is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation, the direct and adjoint methods, and coupled analytic methods for multidisciplinary systems. Each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches. The paper ends with a discussion of current challenges and possib...

Aerodynamic Design Optimization Studies of a Blended-Wing-Body Aircraft

The blended wing body is an aircraft configuration that has the potential to be more efficient than conventional large transport aircraft configurations with the same capability. However, the design of the blended wing is challenging due to the tight coupling between aerodynamic performance, trim, and stability. Other design challenges include the nature and number of the design variables involved, and the transonic flow conditions. To address these issues, a series of aerodynamic shape optimization studies using Reynolds-averaged Navier–Stokes computational fluid dynamics with a Spalart–Allmaras turbulence model is performed. A gradient-based optimization algorithm is used in conjunction with a discrete adjoint method that computes the derivatives of the aerodynamic forces. A total of 273 design variables—twist, airfoil shape, sweep, chord, and span—are considered. The drag coefficient at the cruise condition is minimized subject to lift, trim, static margin, and center plane bending moment constraints. ...

Aerodynamic Shape Optimization Investigations of the Common Research Model Wing Benchmark

Despite considerable research on aerodynamic shape optimization, there is no standard benchmark problem allowing researchers to compare results. This work addresses this issue by solving a series of aerodynamic shape optimization problems based on the Common Research Model wing benchmark case defined by the Aerodynamic Design Optimization Discussion Group. The aerodynamic model solves the Reynolds-averaged Navier–Stokes equations with a Spalart–Allmaras turbulence model. A gradient-based optimization algorithm is used in conjunction with an adjoint method that computes the required derivatives. The drag coefficient is minimized subject to lift, pitching moment, and geometric constraints. A multilevel technique is used to reduce the computational cost of the optimization. A single-point optimization is solved with 720 shape variables using a 28.8-million-cell mesh, reducing the drag by 8.5%. A more realistic design is achieved through a multipoint optimization. Multiple local minima are found when starting...