Optimal design methods involving the solution of an adjoint system of equations are an active area of research in computational fluid dynamics, particularly for aeronautical applications. This paper presents an introduction to the subject, emphasising the simplicity of the ideas when viewed in the context of linear algebra. Detailed discussions also include the extension to p.d.e.'s, the construction of the adjoint p.d.e. and its boundary conditions, and the physical significance of the adjoint solution. The paper concludes with examples of the use of adjoint methods for optimising the design of business jets.

/pdf/an-introduction-to-the-adjoint-approach-to-design-2kv9evoz8u.pdf

An Introduction to the Adjoint Approach to Design

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.

https://hal.archives-ouvertes.fr/hal-01483287/document

The complex-step derivative approximation

A survey is provided of shape parameterization techniques formultidisciplinary optimization, and some emerging ideas are highlighted. The survey focuses on the suitability of available techniques for multidisciplinary applications of complex cone gurations using high-e delity analysis tools such as computational e uid dynamics and computational structural mechanics. The suitability criteria are based on the efe ciency, effectiveness, ease of implementation, and availability of analytical sensitivities for geometry and grids. A section on sensitivity analysis, grid regeneration, and grid deformation techniques is also provided.

Survey of Shape Parameterization Techniques for High-Fidelity Multidisciplinary Shape Optimization

Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows

Grid adaptation for functional outputs: application to two-dimensional inviscid flows

The use of complex variables for determining sensitivity derivatives for turbulent flows is examined. Although a step size parameter is required, the numerical derivatives are not subject to subtractive cancellation errors and, therefore, exhibit true second-order accuracy as the step size is reduced. As a result, this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude. This behavior is in contrast to the use of finite differences, which suffer from inaccuracies due to subtractive cancellation errors. In addition, the complex-variable procedure is easily implemented into existing codes

Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex Variables

Sensitivity analysis for the Navier-Stokes equations on unstructured meshes using complex variables

This manual describes the installation and execution of FUN3D version 13.3, including optional dependent packages. FUN3D is a suite of computational fluid dynamics simulation and design tools that uses mixed-element unstructured grids in a large number of formats, including structured multiblock and overset grid systems. A discretely-exact adjoint solver enables efficient gradient-based design and grid adaptation to reduce estimated discretization error. FUN3D is available with and without a reacting, real-gas capability. This generic gas option is available only for those persons that qualify for its beta release status.

FUN3D Manual: 13.3

Al A A-2001-0596 Recent Improvements in Aerodynamic Design Optimization On Unstructured Meshes

This report describes the research conducted under an interagency collaboration agreement between the Aerospace Systems Directorate of the Air Force Research Laboratory (AFRL/RQ) and the Computational AeroSciences Branch of NASA Langley (NASA LaRC). Both organizations have a long-term goal of developing a modular computational system for coupling fluids and structures to enable both analysis and optimization of aerospace vehicles. Ultimately, the system should support multiple solvers within the fluid and structure domains to allow the best combination for the task at hand, as well as to allow for institutional preferences of specific software components. Towards this goal, the current research was focused on enhancing the existing modal aeroelastic analysis in the NASA FUN3D (Fully-UNstructured three-dimensional CFD (Computational Fluid Dynamics) code) software (Biedron et al. 2018), as well as developing new aeroelastic analysis and optimization capabilities based on a non-linear finite-element method. The methods and enhancements described in this document pertain to FUN3D Version 13.4.

https://ntrs.nasa.gov/api/citations/20180007513/downloads/20180007513.pdf

Eric J. Nielsen

Papers

Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex Variables

Sensitivity analysis for the Navier-Stokes equations on unstructured meshes using complex variables

FUN3D Manual: 13.3

Al A A-2001-0596 Recent Improvements in Aerodynamic Design Optimization On Unstructured Meshes

Sensitivity Analysis for Multidisciplinary Systems (SAMS)