Enabling Automated, Reliable and Efficient Aerodynamic Shape Optimization With Output-Based Adapted Meshes

Simulation-based aerodynamic shape optimization has been greatly pushed forward during the past several decades, largely due to the developments of computational fluid dynamics (CFD), geometry parameterization methods, mesh deformation techniques, sensitivity computation, and numerical optimization algorithms. Effective integration of these components has made aerodynamic shape optimization a highly automated process, requiring less and less human interference. Mesh generation, on the other hand, has become the main overhead of setting up the optimization problem. Obtaining a good computational mesh is essential in CFD simulations for accurate output predictions, which as a result significantly affects the reliability of optimization results. However, this is in general a nontrivial task, heavily relying on the user’s experience, and it can be worse with the emerging high-fidelity requirements or in the design of novel configurations. On the other hand, mesh quality and the associated numerical errors are typically only studied before and after the optimization, leaving the design search path unveiled to numerical errors. This work tackles these issues by integrating an additional component, output-based mesh adaptation, within traditional aerodynamic shape optimizations. First, we develop a more suitable error estimator for optimization problems by taking into account errors in both the objective and constraint outputs. The localized output errors are then used to drive mesh adaptation to achieve the desired accuracy on both the objective and constraint outputs. With the variable fidelity offered by the adaptive meshes, multi-fidelity optimization frameworks are developed to tightly couple mesh adaptation and shape optimization. The objective functional and its sensitivity are first evaluated on an initial coarse mesh, which is then subsequently adapted as the shape optimization proceeds. The effort to set up the optimization is minimal since the initial mesh can be fairly coarse and easy to generate. Meanwhile, the proposed framework saves computational costs by reducing the mesh size at the early stages of the optimization, when the design is far from optimal, and avoiding exhaustive search on low-fidelity meshes when the outputs are inaccurate. To further improve the computational efficiency, we also introduce new methods to accelerate the error estimation and mesh adaptation using machine learning techniques. Surrogate models are developed to predict the localized output error and optimal mesh anisotropy to guide the adaptation. The proposed machine learning approaches demonstrate good performance in two-dimensional test problems, encouraging more study and developments to incorporate them within aerodynamic optimization techniques. Although CFD has been extensively used in aircraft design and optimization, the design automation, reliability, and efficiency are largely limited by the mesh generation process and the fixed-mesh optimization paradigm. With the emerging high-fidelity requirements and the further developments of unconventional configurations, CFD-based optimization has to be made more accurate and more efficient to achieve higher design reliability and lower computational cost. Furthermore, future aerodynamic optimization needs to avoid unnecessary overhead in mesh generation and optimization setup to further automate the design process. The author expects the methods developed in this work to be the keys to enable more automated, reliable, and efficient aerodynamic shape optimization, making CFD-based optimization a more powerful tool in aircraft design.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163034/1/cgderic_1.pd

Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde