Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

Fundamental equations of aerodynamic sensitivity analysis and approximate analysis for the two dimensional thin layer Navier-Stokes equations are reviewed, and special boundary condition considerations necessary to apply these equations to isolated lifting airfoils on 'C' and 'O' meshes are discussed in detail. An efficient strategy which is based on the finite element method and an elastic membrane representation of the computational domain is successfully tested, which circumvents the costly 'brute force' method of obtaining grid sensitivity derivatives, and is also useful in mesh regeneration. The issue of turbulence modeling is addressed in a preliminary study. Aerodynamic shape sensitivity derivatives are efficiently calculated, and their accuracy is validated on two viscous test problems, including: (1) internal flow through a double throat nozzle, and (2) external flow over a NACA 4-digit airfoil. An automated aerodynamic design optimization strategy is outlined which includes the use of a design optimization program, an aerodynamic flow analysis code, an aerodynamic sensitivity and approximate analysis code, and a mesh regeneration and grid sensitivity analysis code. Application of the optimization methodology to the two test problems in each case resulted in a new design having a significantly improved performance in the aerodynamic response of interest

Using Automatic Differentiation for Adjoint CFD Code Development

This paper addresses the concerns of CFD code developers who are facing the task of creating a discrete adjoint CFD code for design optimisation. It discusses how the development of such a code can be greatly eased through the selective use of Automatic Differentiation, and how the software development can be subjected to a sequence of checks to ensure the correctness of the final software

A coupled discrete adjoint method for optimal design with dynamic non-linear fluid structure interactions

Incorporating high-fidelity analysis methods in multidisciplinary design optimization necessitates efficient sensitivity evaluation, which is particularly important for time-accurate problems. This thesis presents a new discrete adjoint formulation suitable for fully coupled, non-linear, dynamic FSI problems. The solution includes time-dependent adjoint variables that arise from grid motion and chosen time integration methods for both the fluid and structural domains. Implemented as a generic multizone discrete adjoint solver for time-accurate analysis in the open-source multiphysics solver SU2, this provides a flexible framework for a wide range of applications. Design optimization of aerodynamic structures need accurate characterization of the coupled fluid-structure interactions (FSI). Incorporating high-fidelity analysis methods in the multidisciplinary design optimization (MDO) necessitates efficient sensitivity evaluation, which is particularly important for time-accurate problems. Adjoint methods are well established for sensitivity analysis when large number of design variables are needed. The use of discrete adjoint method through algorithmic differentiation enables the evaluation of sensitivities using an approximation of the Jacobian of the coupled problem, thus enabling this approach to be applied for multidisciplinary analysis. This thesis presents a new discrete adjoint formulation suitable for fully coupled, non-linear, dynamic FSI problems. A partitioned approach is considered with finite volume for the fluid and finite elements for the solid domains. The solution includes the time-dependent adjoint variables that arise from the grid motion and chosen time integration methods for both the fluid and structural domains. Implemented as a generic multizone discrete adjoint solver for timeaccurate analysis in the open-source multiphysics solver SU2, this provides a flexible framework for a wide range of applications. The partitioned FSI solver approach has been leveraged to extend the dynamic FSI capabilities to low speed flows through the introduction of a densitybased unsteady incompressible flow solver. The developed methodology and implementation are demonstrated using a range of numerical test cases. Optimal design for steady, coupled FSI problems are firstly presented before moving to the building blocks of dynamic coupled problems using single domain analysis, for both structural and fluid domains in turn. The new unsteady incompressible fluid solver, for both the primal and adjoint analysis, are verified against a range of well-known benchmark test cases, including problems with grid motion. Finally, applications of coupled dynamic problems are presented to verify both the unsteady incompressible solver for FSI as well as the successful verification of the discrete adjoint sensitivities for the transient response of a transonic compliant airfoil for a variety of both aerodynamic and structural objective functions.Open Acces

Investigation of Adjoint Based Shape Optimization Techniques in NASCART-GT using Automatic Reverse Differentiation

Automated shape optimization involves making suitable modifications to a geometry that can lead to significant improvements in aerodynamic performance. Currently available mid-fdelity Aerodynamic Optimizers cannot be utilized in the late stages of the design process for performing minor, but consequential, tweaks in geometry. Automated shape optimization involves making suitable modifications to a geometry that can lead to significant improvements in aerodynamic performance. Currently available mid-fidelity Aerodynamic Optimizers cannot be utilized in the late stages of the design process for performing minor, but consequential, tweaks in geometry. High-fidelity shape optimization techniques are explored which, even though computationally demanding, are invaluable since they can account for realistic effects like turbulence and viscocity. The high computational costs associated with the optimization have been avoided by using an indirect optimization approach, which was used to dcouple the effect of the flow field variables on the gradients involved. The main challenge while performing the optimization was to maintain low sensitivity to the number of input design variables. This necessitated the use of Reverse Automatic differentiation tools to generate the gradient. All efforts have been made to keep computational costs to a minimum, thereby enabling hi-fidelity optimization to be used even in the initial design stages. A preliminary roadmap has been laid out for an initial implementation of optimization algorithms using the adjoint approach, into the high fidelity CFD code NASCART-GT.High-fidelity shape optimization techniques are explored which, even though computationally demanding, are invaluable since they can account for realistic effects like turbulence and viscocity. The high computational costs associated with the optimization have been avoided by using an indirect optimization approach, which was used to dcouple the effect of the flow field variables on the gradients involved. The main challenge while performing the optimization was to maintain low sensitivity to the number of input design variables. This necessitated the use of Reverse Automatic differentiation tools to generate the gradient. All efforts have been made to keep computational costs to a minimum, thereby enabling hi-fidelity optimization to be used even in the initial design stages. A preliminary roadmap has been laid out for an initial implementation of optimization algorithms using the adjoint approach, into the high fidelity CFD code NASCART-GT.Ruffin, Stephen - Faculty Mentor ; Feron, Eric - Committee Member/Second Reader ; Sankar, Lakshmi - Committee Member/Second Reade

Combining Parameterizations, Sobolev Methods and Shape Hessian Approximations for Aerodynamic Design Optimization

Aerodynamic design optimization, considered in this thesis, is a large and complex area spanning different disciplines from mathematics to engineering. To perform optimizations on industrially relevant test cases, various algorithms and techniques have been proposed throughout the literature, including the Sobolev smoothing of gradients. This thesis combines the Sobolev methodology for PDE constrained flow problems with the parameterization of the computational grid and interprets the resulting matrix as an approximation of the reduced shape Hessian. Traditionally, Sobolev gradient methods help prevent a loss of regularity and reduce high-frequency noise in the derivative calculation. Such a reinterpretation of the gradient in a different Hilbert space can be seen as a shape Hessian approximation. In the past, such approaches have been formulated in a non-parametric setting, while industrially relevant applications usually have a parameterized setting. In this thesis, the presence of a design parameterization for the shape description is explicitly considered. This research aims to demonstrate how a combination of Sobolev methods and parameterization can be done successfully, using a novel mathematical result based on the generalized Faà di Bruno formula. Such a formulation can yield benefits even if a smooth parameterization is already used. The results obtained allow for the formulation of an efficient and flexible optimization strategy, which can incorporate the Sobolev smoothing procedure for test cases where a parameterization describes the shape, e.g., a CAD model, and where additional constraints on the geometry and the flow are to be considered. Furthermore, the algorithm is also extended to One Shot optimization methods. One Shot algorithms are a tool for simultaneous analysis and design when dealing with inexact flow and adjoint solutions in a PDE constrained optimization. The proposed parameterized Sobolev smoothing approach is especially beneficial in such a setting to ensure a fast and robust convergence towards an optimal design. Key features of the implementation of the algorithms developed herein are pointed out, including the construction of the Laplace-Beltrami operator via finite elements and an efficient evaluation of the parameterization Jacobian using algorithmic differentiation. The newly derived algorithms are applied to relevant test cases featuring drag minimization problems, particularly for three-dimensional flows with turbulent RANS equations. These problems include additional constraints on the flow, e.g., constant lift, and the geometry, e.g., minimal thickness. The Sobolev smoothing combined with the parameterization is applied in classical and One Shot optimization settings and is compared to other traditional optimization algorithms. The numerical results show a performance improvement in runtime for the new combined algorithm over a classical Quasi-Newton scheme

Towards Reduced-order Model Accelerated Optimization for Aerodynamic Design

The adoption of mathematically formal simulation-based optimization approaches within aerodynamic design depends upon a delicate balance of affordability and accessibility. Techniques are needed to accelerate the simulation-based optimization process, but they must remain approachable enough for the implementation time to not eliminate the cost savings or act as a barrier to adoption. This dissertation introduces a reduced-order model technique for accelerating fixed-point iterative solvers (e.g. such as those employed to solve primal equations, sensitivity equations, design equations, and their combination). The reduced-order model-based acceleration technique collects snapshots of early iteration (pre-convergent) solutions and residuals and then uses them to project to significantly more accurate solutions, i.e. smaller residual. The technique can be combined with other convergence schemes like multigrid and adaptive timestepping. The technique is generalizable and in this work is demonstrated to accelerate steady and unsteady flow solutions; continuous and discrete adjoint sensitivity solutions; and one-shot design optimization solutions. This final application, reduced-order model accelerated one-shot optimization approach, in particular represents a step towards more efficient aerodynamic design optimization. Through this series of applications, different basis vectors were considered and best practices for snapshot collection procedures were outlined. The major outcome of this dissertation is the development and demonstration of this reduced-order model acceleration technique. This work includes the first application of the reduced-order model-based acceleration method to an explicit one-shot iterative optimization process

Adjoint-based aerodynamic shape optimization on unstructured meshes

In this paper, the exact discrete adjoint of an unstructured finite-volume formulation of the Euler equations in two dimensions is derived and implemented. The adjoint equations are solved with the same implicit scheme as used for the flow equations. The scheme is modified to efficiently account for multiple functionals simultaneously. An optimization framework, which couples an analytical shape parameterization to the flow/adjoint solver and to algorithms for constrained optimization, is tested on airfoil design cases involving transonic as well as supersonic flows. The effect of some approximations in the discrete adjoint, which aim at reducing the complexity of the implementation, is shown in terms of optimization results rather than only in terms of gradient accuracy. The shape-optimization method appears to be very efficient and robust

Component-based Geometry Manipulation for Aerodynamic Shape Optimization with Overset Meshes

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143082/1/6.2017-3327.pd