Nicolas R. Gauger

Three-Dimensional Large-Scale Aerodynamic Shape Optimization Based on Shape Calculus

Large-scale three-dimensional aerodynamic shape optimization based on the compressible Euler equations is considered. Shape calculus is used to derive an exact surface formulation of the gradients, enabling the computation of shape gradient information for each surface mesh node without having to calculate further mesh sensitivities. Special attention is paid to the applicability to large-scale three dimensional problems like the optimization of an Onera M6 wing or a complete blended-wing–body aircraft. The actual optimization is conducted in a one-shot fashion, in which the tangential Laplace operator is used as a Hessian approximation, thereby also preserving the regularity of the shape.

Single-step One-shot Aerodynamic Shape Optimization

In this paper we consider the shape optimization of a transonic airfoil whose aerodynamic properties are calculated by a structured Euler solver. The optimization strategy is based on a one-shot technique in which pseudo time-steps of the primal and the adjoint solver are iterated simultaneously with design corrections done on the airfoil geometry. The adjoint solver which calculates the necessary sensitivities is based on discrete adjoints and derived by using reverse mode of automatic differentiation. A new preconditioner which is derived by considering an augmented Lagrangian formulation of the optimization problem is employed in order to achieve bounded retardation of the overall optimization process. A design example of drag minimization for an RAE2822 airfoil under transonic flight conditions is included.

Automatic Differentiation of an Entire Design Chain for Aerodynamic Shape Optimization

Detailed numerical shape optimization will play a strategic role for future aircraft design. It offers the possibility of designing or improving aircraft components with respect to a pre-specified figure of merit. Optimization methods based on exact derivatives of the cost function with respect to the design variables still suffer from the high computational costs if many design variables are used. However, these gradients can be efficiently obtained by the solution of so-called adjoint flow equations. Furthermore, it is possible to derive these adjoint solvers in an automated fashion by the use of so-called automatic differentiation (AD) tools. In the present paper the efficient and automated differentiation of an entire design chain, including the flow solver, is presented. As test case for AD-generated adjoint sensitivity calculations an inviscid RAE2822 airfoil is chosen under transonic flight conditions for drag reduction.

Sensitivity analysis and optimization of sub-wavelength optical gratings using adjoints.

Numerical optimization of photonic devices is often limited by a large design space the finite-differences gradient method requires as many electric field computations as there are design parameters. Adjoint-based optimization can deliver the same gradients with only two electric field computations. Here, we derive the relevant adjoint formalism and illustrate its application for a waveguide slab, and for the design of optical sub-wavelength gratings.

Automated Extension of Fixed Point PDE Solvers for Optimal Design with Bounded Retardation

We study PDE-constrained optimization problems where the state equation is solved by a pseudo-time stepping or fixed point iteration. We present a technique that improves primal, dual feasibility and optimality simultaneously in each iteration step, thus coupling state and adjoint iteration and control/design update. Our goal is to obtain bounded retardation of this coupled iteration compared to the original one for the state, since the latter in many cases has only a Q-factor close to one. For this purpose and based on a doubly augmented Lagrangian, which can be shown to be an exact penalty function, we discuss in detail the choice of an appropriate control or design space preconditioner, discuss implementation issues and present a convergence analysis. We show numerical examples, among them applications from shape design in fluid mechanics and parameter optimization in a climate model.

Development of a Consistent Discrete Adjoint Solver in an Evolving Aerodynamic Design Framework

Typically the development of adjoint solvers for the use in aerodynamic design is challenging. In this paper we will give an update on the development of a discrete adjoint solver that enables the computation of consistent gradients within the open-source multi-physics framework SU2. Due to the use of advanced programming techniques like Expression Templates and the application of Algorithmic Differentiation we obtain an automatic adaption to modifications and extensions of the flow/state solver while maintaining robustness and efficiency.

Adjoint Approaches For Optimal Flow Control

In many engineering applications, aerodynamic behaviour is significantly influenced by flow separation. A promising concept for improving the aerodynamic design is to manipulate the separated flow using active flow control. This is often realised by blowing and suction, whereby gradient-based optimisation methods can help to find an optimal set of excitation parameters. In the present paper, three methods for calculating the gradient are compared: Finite Differences, a continuous adjoint approach and a discrete adjoint method based on Automatic Differentiation (AD). The methods have been applied to the flow around a rotating cylinder at Reynolds numbers of Re = 100 and Re = 5000 to calculate the derivative of the drag coefficient with respect to the revolution rate of the cylinder. The results of these computations are used for a detailed comparison of the methods in terms of consistency, numerical efficiency and accuracy with special emphasis on the treatment of turbulence to demonstrate the superiority of the discrete adjoint approach.

Optimal Control of Unsteady Flows Using Discrete Adjoints

We present the development of a discrete adjoint approach for the optimal control of viscous ows, governed by unsteady incompressible Reynolds Averaged Navier Stokes equations. The adjoint solver is developed by applying the automatic di erentiation (AD) techniques in reverse mode of di erentiation. The unsteady adjoints usually require the storage of complete ow history during the forward-in-time integration of the primal equations, which is then used while solving the adjoint equations in backward-in-time integration. For large scale applications, the memory requirements for storing the ow solutions can become prohibitively expensive. To reduce the excessive memory demands, the binomial checkpointing strategy has been employed. Numerical results are presented for the test cases of optimal active ow control around a rotating cylinder and a NACA4412 airfoil to validate the automatic di erentiation generated derivative codes. The sensitivities based on forward and adjoint mode AD codes are compared with the values obtained from nite di erences.

Differentiating Fixed Point Iterations with ADOL-C: Gradient Calculation for Fluid Dynamics

The reverse mode of automatic differentiation allows the computation of gradients at a temporal complexity that is only a small multiple of the temporal complexity to evaluate the function itself. However, the memory requirement of the reverse mode in its basic form is proportional to the operation count of the function to be differentiated. For iterative processes consisting of iterations with uniform complexity this means that the memory requirement of the reverse mode grows linearly with the number of iterations. For fixed point iterations this is not efficient, since any structure of the problem is neglected.

One-shot methods in function space for PDE-constrained optimal control problems

We state and analyse one-shot methods in function space for the optimal control of nonlinear partial differential equations (PDEs) that can be formulated in terms of a fixed-point operator. A general convergence theorem is proved by generalizing the previously obtained results in finite dimensions. As application examples we consider two nonlinear elliptic model problems: the stationary solid fuel ignition model and the stationary viscous Burgers equation. For these problems we present a more detailed convergence analysis of the method. The resulting algorithms are computationally implemented in combination with an adaptive mesh refinement strategy, which leads to an improvement in the performance of the one-shot approach.