Aerodynamic shape optimization by multi-fidelity modeling and manifold mapping

Aerodynamic shape optimization (ASO) is important in contemporary engineering design of complex systems such as aircraft and wind turbines. The use of high-fidelity partial differential equation (PDE) simulations within the design process is becoming the standard. However, the overall computation cost of the ASO problem can be very high when considering the following key challenges: (1) time-consuming PDE simulations, (2) large number of design variables, and (3) conventional optimization require many system evaluations. Combined these form an optimization problem which may be prohibitive to solve, even when using high performance computing (HPC) systems. In this work, 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. Only one high-fidelity PDE simulation is required per design iteration. The version of MM utilized here does not require gradient information. The proposed method is validated and characterized by applying it to several benchmark ASO problems, including lift-constrained airfoil drag minimization in inviscid and viscous transonic flows, and comparing the results with state of the art techniques. 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. In other words, the MM algorithm was an order of magnitude faster than the gradient-based search with adjoint sensitivities in this case. For the viscous case, MM yields an optimized shape using less than 300 equivalent high-fidelity evaluations, taking approximately 80 hours on the HPC. In this case, the direct algorithm was not able to reach a comparably efficient shape. MM is able to handle vector-valued design problems efficiently. This was demonstrated on a multipoint design problem as well as on an inverse design problem. The multipoint design showed that the optimized airfoil outperformed that original airfoil in terms of flight conditions and robustness in multiple cruise conditions. In the inverse design case, MM needed an order of magnitude less equivalent high-fidelity model evaluations than a derivative-free algorithm

Multi-Fidelity Methods for Optimization: A Survey

Real-world black-box optimization often involves time-consuming or costly experiments and simulations. Multi-fidelity optimization (MFO) stands out as a cost-effective strategy that balances high-fidelity accuracy with computational efficiency through a hierarchical fidelity approach. This survey presents a systematic exploration of MFO, underpinned by a novel text mining framework based on a pre-trained language model. We delve deep into the foundational principles and methodologies of MFO, focusing on three core components -- multi-fidelity surrogate models, fidelity management strategies, and optimization techniques. Additionally, this survey highlights the diverse applications of MFO across several key domains, including machine learning, engineering design optimization, and scientific discovery, showcasing the adaptability and effectiveness of MFO in tackling complex computational challenges. Furthermore, we also envision several emerging challenges and prospects in the MFO landscape, spanning scalability, the composition of lower fidelities, and the integration of human-in-the-loop approaches at the algorithmic level. We also address critical issues related to benchmarking and the advancement of open science within the MFO community. Overall, this survey aims to catalyze further research and foster collaborations in MFO, setting the stage for future innovations and breakthroughs in the field.Comment: 47 pages, 9 figure