Mechanical Engineering
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Item Analyzing Inverse Design Problems from a Topological Perspective(2024) Chen, Qiuyi; Fuge, Mark; Mechanical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Inverse design (ID) problems are inverse problems that aim to rapidly retrieve the subset of valid designs having the desired performances and properties under the given conditions. In practice, this can be solved by training generative models to approximate and sample the posterior distributions of designs. However, little has been done to understand their mechanisms and limitations from a theoretical perspective. This dissertation leverages theoretical tools from general and differential topology to answer these three questions of inverse design: what does a set of valid designs look like? How helpful are the data-driven generative models for retrieving the desired designs from this set? What topological properties affect the subset of desired designs? The dissertation proceeds by dismantling inverse (design) problems into two major subjects: that is, the representing and probing of a given set of valid designs (or data), and the retrieval of the desired designs (or data) from this given set. It draws inspiration from topology and geometry to investigate them and makes the main contributions below: 1. Chapter 3 details a novel representation learning method called Least Volume, which has properties similar to nonlinear PCA for representing datasets. It can minimize the representation's dimension automatically and, as shown in Chapter 4, conducts contrastive learning when applied to labeled datasets. 2. Two conditional generative models are developed to generate performant 2-D airfoils and 3-D heat sinks in Chapter 5 and 6 respectively. They can produce realistic designs to warm-start further optimization, with the relevant chapters detailing their acceleration effects. 3. Lastly, Chapter 7 describes how to use Least volume to solve high-dimensional inverse problems efficiently. Specifically, using examples from physic system identification, the chapter uncovers the correlation between the inverse problem's uncertainty and its intrinsic dimensions.Item Denoising the Design Space: Diffusion Models for Accelerated Airfoil Shape Optimization(2024) Diniz, Cashen; Fuge, Mark D; Mechanical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Generative models offer the possibility to accelerate and potentially substitute parts of the often expensive traditional design optimization process. We present Aero-DDM, a novel application of a latent denoising diffusion model (DDM) capable of generating airfoil geometries conditioned on flow parameters and an area constraint. Additionally, we create a novel, diverse dataset of optimized airfoil designs that better reflects a realistic design space than has been done in previous work. Aero-DDM is applied to this dataset, and key metrics are assessed both statistically and with an open-source computational fluid dynamics (CFD) solver to determine the performance of the generated designs. We compare our approach to an optimal transport GAN, and demonstrate that our model can generate designs with superior performance statistically, in aerodynamic benchmarks, and in warm-start scenarios. We also extend our diffusion model approach, and demonstrate that the number of steps required for inference can be reduced by as much as ~86%, compared to an optimized version of the baseline inference process, without meaningful degradation in design quality, simply by using the initial design to start the denoising process.