Full Domain Analysis in Fluid Dynamics

TL;DR

The paper proposes Full Domain Analysis (FDA) as a framework to comprehensively map and study the full solution space $\,\mathbf{S}$ in expensive CFD problems, using encodings, diverse search paradigms, and surrogate-assisted methods to explore morphology–flow relationships. It surveys encoding strategies (Direct, Indirect, Latent) and three search paradigms (Pareto, Multimodal, Quality Diversity), and emphasizes efficiency via surrogates, replacement by latent models, and GPU-accelerated CFD. A demonstrative 2D flow domain shows how to build and analyze large archives of designs, employing a CVAE for generation and a latent walk to relate shape features to flow metrics such as $u_{max}$ and $\mathcal{E}$. The work demonstrates that combining robust encodings, diverse search, and surrogate-assisted evaluation enables rapid, interactive exploration of design spaces, aiding engineers in understanding trade-offs, anticipating risks, and iterating toward robust, innovative solutions. It also outlines avenues for future improvements in model explainability, disentanglement of latent factors, and integration with multi-fidelity modeling to further accelerate discovery in computational physics domains.

Abstract

Novel techniques in evolutionary optimization, simulation and machine learning allow for a broad analysis of domains like fluid dynamics, in which computation is expensive and flow behavior is complex. Under the term of full domain analysis we understand the ability to efficiently determine the full space of solutions in a problem domain, and analyze the behavior of those solutions in an accessible and interactive manner. The goal of full domain analysis is to deepen our understanding of domains by generating many examples of flow, their diversification, optimization and analysis. We define a formal model for full domain analysis, its current state of the art, and requirements of subcomponents. Finally, an example is given to show what we can learn by using full domain analysis. Full domain analysis, rooted in optimization and machine learning, can be a helpful tool in understanding complex systems in computational physics and beyond.

Figures (12)

• Figure 1: User process perspective on . After the user defines the domain through available initial parameters, constraints and objectives, the prior (1), a large variety of designs is automatically generated (2). The user examines what makes designs perform well (3). They then select promising design regions which allows to update the initial domain definition (4). The next iteration of the process zooms in on the user's region of interest.
• Figure 2: While search and optimization takes place in which we call the parameter space $\mathbf{X}$, through the encoding's expression and its in-situ simulation, only a part of the solution space $\mathbf{S}$ can be reached, the reachable manifold $\mathbf{R}$. This manifold might include invalid solutions (necessitating constraints on the parameter space). The goal is to maximize $\mathbf{R}$ w. r. t. the valid subspace $\mathbf{V}$.
• Figure 3: Three main categories of shape encodings to produce solutions in $\mathbf{S}$. Direct, parameterized encodings use a manually defined decoder that determines spline shapes. Indirect encodings search the shapes indirectly, by performing search on the functional structure of the decoder. The decoder determines which pixel or voxel in a discretized solution is filled. Data-driven latent-generative approaches use pre-trained trained generative models that compress a set of prior examples to a low-dimensional latent space, which serves as a search space $\mathbf{X}$.
• Figure 4: Heterogeneous fitness landscapes often contain clusters of varying sizes, making the definition of "local optimum" in terms of a threshold distance value $\epsilon$ indeterminable. $\epsilon_1$ and $\epsilon_2$ vary.
• Figure 5: Multi-solution optimization. Multi-objective optimization (a) finds a Pareto front of trade-off solutions. Solutions are added to the front if they dominate neighboring solutions in at least one objective. In multimodal optimization (b), solutions are selected through local competition in the parameter space. Quality diversity (c) searches in parameter space, but local competition takes place in a low-dimensional archive defined by characteristics of their morphology or behavior.