Leifur T. Leifsson

Application of Multifidelity Optimization Techniques to Benchmark Aerodynamic Design Problems

Two-dimensional benchmark cases involving lift-constrained drag minimization in inviscid and viscous transonic flows are solved using derivative-free multi-fidelity optimization algorithms (space mapping and manifold mapping) and are compared with direct gradient-based optimization algorithms using adjoint sensitivities and trust regions. With 8 B-spline design variables, the multi-fidelity algorithms yield optimized shapes comparable to the shapes obtained by the direct algorithms but at a fraction of the cost. In particular for the inviscid case, the multi-fidelity algorithms needed less than 150 equivalent high-fidelity model evaluations (only flow solutions) taking approximately 460 minutes on a HPC with 32 processors, whereas the direct algorithm needed 391 high-fidelity model evaluations (flow and adjoint) taking approximately 4,494 minutes on the same HPC. For the viscous case, the multi-fidelity algorithms yield an optimized shape using less than 125 equivalent high-fidelity evaluations taking approximately 17.4 hours on the HPC. The direct algorithm was unsuccessful in optimizing the baseline shape in this case. A simple variation of surrogate-based optimization, the sequential approximate optimization (SAO), is utilized to optimize the twist distribution of a rectangular unswept wing in inviscid flow. Using 3 Bspline design variables, the SAO algorithm is able to obtain an optimized design with a nearelliptic section lift distribution. The total optimization cost is 22 high-fidelity model evaluations or approximately 42.5 hours on a HPC with 32 processors.

Surrogate-Based Airfoil Design with Multi-Level Optimization and Adjoint Sensitivity

Robust and computationally efficient airfoil shape optimization algorithm is presented. Our technique enhances a recently introduced multi-level optimization (MLO) algorithm with adjoint sensitivity. Adjoint-enhanced MLO exploits a set of computational fluid dynamics (CFD) models of increasing discretization density that are sequentially optimized with the optimal design of the “coarser” model being the initial design for the “finer” one. Transition between the models of different fidelities is governed by suitably selected termination criteria, more relaxed at the initial stages, and stricter towards the end of the optimization process. Exploitation of variable-fidelity models allows for taking larger steps in the design space at a lower CPU cost, leading to an improved efficiency when compared to gradient-based direct optimization of the high-fidelity model with adjoints as demonstrated by several test cases of transonic airfoils. In particular, the results show that up to a 43% reduction in the optimization cost can be achieved with MLO when compared with the direct approach.

Multi-Fidelity Aerodynamic Shape Optimization Using Manifold Mapping

A computationally efficient optimization algorithm for aerodynamic design is presented. In our approach, direct optimization of a computationally expensive model is replaced by an iterative updating and re-optimization of a fast physics-based replacement model, following the surrogate-based optimization paradigm. The surrogate is constructed using a low-fidelity model which is corrected using manifold mapping (MM) to become a reliable representation of the high-fidelity one during the optimization process. MM exploits the high-fidelity simulation data accumulated in the course of the optimization run gradually improving the generalization capability of the surrogate. The version of MM utilized here does not require gradient information. The method is applied to lift-constrained airfoil drag minimization in two-dimensional inviscid and viscous transonic flows. MM yielded optimized shapes, with 8 B-spline design variables, that are comparable to the shapes obtained by direct optimization algorithms equipped with adjoint sensitivities and trust regions. In the inviscid benchmark case, MM needed less than 150 equivalent high-fidelity model evaluations (only flow solutions), or approximately 460 minutes on a HPC with 32 processors, whereas the direct algorithm needed 391 high-fidelity model evaluations (flow and adjoint), or approximately 4,494 minutes on the same HPC. For the viscous case, MM yields an optimized shape using less than 125 equivalent high-fidelity evaluations, taking approximately 17.4 hours on the HPC. The direct algorithm was unsuccessful in optimizing the baseline shape in this case.

Direct and Surrogate-Based Optimization of Dual-Rotor Wind Turbines

Dual-rotor wind turbines (DRWT) may offer better energy efficiency over their singlerotor counterparts. The design and analysis of DRWT requires, among other, the use of computational fluid dynamics models. These models can be, depending on their formulation, computationally heavy. Numerous simulations are then required during the design process, and this may render the overall computational cost to be prohibitive. This paper investigates and compares several optimization techniques for the design of DRWTs. In particular, we solve the DRWT fluid flow using the Reynolds-Averaged Navier-Stokes equations with a two-equation turbulence model on an axisymmetric mesh, and consider three design approaches: the traditional parametric sweep where the design variables are varied and the responses examined, direct optimization with a derivative-free algorithm, and surrogate-based optimization (SBO) using data-driven surrogates. The approaches are applied to test cases involving two and three design variables. The results show that the same optimized designs are obtained with all the approaches. However, going from the two parameter case to the three parameter one, the effort of setting up, running, and analyzing the results increases significantly with the parametric sweep approach. The optimization techniques are much easier to use and deliver the results with lower computational cost, where the SBO algorithm outperforms the direct approach.

Multi-objective Airfoil Design Using Variable-Fidelity CFD Simulations and Response Surface Surrogates

In this work, a computationally efficient procedure for multi-objective design optimization of transonic airfoil shapes is presented. The proposed approach utilizes the multi-objective evolutionary algorithm (MOEA) that works with a fast surrogate model of an airfoil under design, obtained with kriging interpolation of low-fidelity CFD airfoil simulations. The initial Pareto front generated by multi-objective optimization of the surrogate using MOEA can be iteratively refined by local enhancements of the surrogate model. The latter are realized with space mapping response correction based on limited number of high-fidelity CFD training points allocated along the initial Pareto front. The proposed method allows us to obtain—at a low computational cost—a set of airfoil geometries representing trade-offs between the lift and drag coefficients. Our approach is illustrated using an example design of a transonic airfoil.

Physics-Based Multi-Fidelity Surrogate Modeling with Entropy-Based Availability Methods

Thermodynamic availability of an engineering system is described in terms of entropy generation and is a common parameter applicable to all processes and subsystems within the system. Availability methods based on this metric can be employed for the characterization of aerospace vehicle performance and mission analysis. Models of varying complexity and computational expense are typically used in vehicle design. This paper introduces an approach to construct cheap and accurate surrogate models for availability methods and analysis. The proposed technique is physics-based and aligns entropy-based metrics of computationally cheap low-fidelity models with the expensive high-fidelity models using implicit space mapping. The resulting surrogates are fast and have good generalization capabilities. The approach is applied to the compressible, viscous flow in a convergentdivergent nozzle modeled with an analytic model and a high-fidelity computational fluid dynamics model. The models are aligned in terms of the governing flow variables by aligning the entropy generation across the normal shock. The proposed approach is compared with standard correction/enhancement techniques.

Trawl-door design optimization by local surrogate models

Trawl-doors have a large influence on the fuel consumption of fishing vessels. Design and optimization of trawl-doors using computational models are a key factor in minimizing the fuel consumption. This paper presents an optimization algorithm for the shape design of trawl-door shapes using computational fluid dynamic (CFD) models. Accurate CFD models are computationally expensive. Therefore, the direct use of traditional optimization algorithms, which often require a large number of evaluations, may prohibitive. The proposed approach is iterative and uses low-order local response surface approximation models of the expensive CFD model, constructed in each iteration, to reduce the number of evaluations. The algorithm is applied to the design of a two-element trawl-door (slat and airfoil), involving four design variables controlling the angle of attack and the slat position and orientation. The results show that a satisfactory design can be obtained at the cost of a few iterations of the algorithm.

Automated Low-fidelity Model Selection for CFD-based Aerodynamic Shape Optimization

Computational fluid dynamics (CFD) simulations are a fundamental tool in aerodynamic design. Unfortunately, accurate, high-fidelity CFD models may be computationally too expensive to conduct the design using numerical optimization procedures. Recently, variable-fidelity optimization algorithms have attracted attention for their ability to reduce high CPU-cost related to the design process solely based on accurate CFD models. Lowfidelity simulation models are the most critical components of such algorithms. They normally employ the same CFD solver as the high-fidelity model but with reduced discretization density and reduced number of flow solver iterations. Typically, the selection of the appropriate model parameters has only been guided by the designer experience. In this work, an automated low-fidelity model selection technique is presented. By defining the model setup task as a constrained nonlinear optimization problem, suitable grid and flow solver parameters are obtained. Our approach is compared to two conventional methods of generating a family of variable-fidelity models. Comparison of the standard and the proposed approach is carried out in the context of aerodynamic design of a transonic airfoil using a multi-level optimization algorithm. The results obtained for several test cases indicate that the automated model generation may lead to significant computational savings of the CFD-based airfoil design process.

Rapid Multi-Objective Aerodynamic Design Using Co-Kriging and Space Mapping

In this paper, a procedure for computationally feasible multi-objective design optimization of aerodynamic surfaces is presented. Our approach exploits multi-fidelity aerodynamics models as well as a multi-objective evolutionary algorithm (MOEA). For the sake of cost reduction, the initial Pareto front is obtained by optimizing a fast kriging surrogate model using MOEA. The surrogate is constructed from sampled low-fidelity model which is pre-conditioned using high-fidelity model data and space mapping. The surrogate is then iteratively refined by enhancing it using high-fidelity model data points sampled along the Pareto set using co-kriging. The process is continued until the Pareto front representation produced by the surrogate aligns with the high-fidelity verification samples. The proposed method allows us to obtain—at a low computational cost—a set of aerodynamic geometries representing trade-offs between the figures of merit. Our approach is illustrated on the design of airfoil shapes in transonic flow at constant lift and obtaining the Pareto front for the drag and pitching moment coefficients.

Simulation-Driven Aerodynamic Shape Optimization with Automated Low-Fidelity Model Setup

Accurate high-fidelity Computational Fluid Dynamics (CFD) models may be computationally too expensive for simulation-driven design optimization. Variable-fidelity optimization algorithms have been utilized to reduce high CPU-cost related to the design process solely based on accurate CFD models. The most critical components of such algorithms are the low-fidelity models. Typically, the low-fidelity models employ the same CFD solver as the high-fidelity one, but with reduced discretization density and reduced number of flow solver iterations. The performance of the optimization algorithm strongly depends on the quality of the low-fidelity models. The low-fidelity model grid setup has been based on hands-on parametric studies. In this work, an automated low-fidelity CFD model setup technique is developed. The model setup task is defined as a constrained nonlinear optimization problem and suitable grid and flow solver parameters are obtained numerically. Comparison of the standard and the proposed approach is carried out in the context of aerodynamic design of transonic airfoils. Two variable-fidelity optimization algorithms are used in the study. One algorithm is based on a single corrected low-fidelity CFD model and the other utilizes a family of such models. The results suggest that the automated model generation may lead to significant computational savings of the CFDbased aerodynamic design process.