SCI Publications

Deep neural operators, such as DeepONets, have changed the paradigm in high-dimensional nonlinear regression from function regression to (differential) operator regression, paving the way for significant changes in computational engineering applications. Here, we investigate the use of DeepONets to infer flow fields around unseen airfoils with the aim of shape optimization, an important design problem in aerodynamics that typically taxes computational resources heavily. We present results which display little to no degradation in prediction accuracy, while reducing the online optimization cost by orders of magnitude. We consider NACA airfoils as a test case for our proposed approach, as their shape can be easily defined by the four-digit parametrization. We successfully optimize the constrained NACA four-digit problem with respect to maximizing the lift-to-drag ratio and validate all results by comparing them to a high-order CFD solver. We find that DeepONets have low generalization error, making them ideal for generating solutions of unseen shapes. Specifically, pressure, density, and velocity fields are accurately inferred at a fraction of a second, hence enabling the use of general objective functions beyond the maximization of the lift-to-drag ratio considered in the current work.

Deep neural operators, such as DeepONet, have changed the paradigm in high-dimensional nonlinear regression, paving the way for significant generalization and speed-up in computational engineering applications. Here, we investigate the use of DeepONet to infer flow fields around unseen airfoils with the aim of shape constrained optimization, an important design problem in aerodynamics that typically taxes computational resources heavily. We present results that display little to no degradation in prediction accuracy while reducing the online optimization cost by orders of magnitude. We consider NACA airfoils as a test case for our proposed approach, as the four-digit parameterization can easily define their shape. We successfully optimize the constrained NACA four-digit problem with respect to maximizing the lift-to-drag ratio and validate all results by comparing them to a high-order CFD solver. We find that DeepONets have a low generalization error, making them ideal for generating solutions of unseen shapes. Specifically, pressure, density, and velocity fields are accurately inferred at a fraction of a second, hence enabling the use of general objective functions beyond the maximization of the lift-to-drag ratio considered in the current work. Finally, we validate the ability of DeepONet to handle a complex 3D waverider geometry at hypersonic flight by inferring shear stress and heat flux distributions on its surface at unseen angles of attack. The main contribution of this paper is a modular integrated design framework that uses an over-parametrized neural operator as a surrogate model with good generalizability coupled seamlessly with multiple optimization solvers in a plug-and-play mode.

Computational fluid dynamics (CFD) is known for producing high-dimensional spatiotemporal data. Recent advances in machine learning (ML) have introduced a myriad of techniques for extracting physical information from CFD. Identifying an optimal set of coordinates for representing the data in a low-dimensional embedding is a crucial first step toward data-driven reduced-order modeling and other ML tasks. This is usually done via principal component analysis (PCA), which gives an optimal linear approximation. However, fluid flows are often complex and have nonlinear structures, which cannot be discovered or efficiently represented by PCA. Several unsupervised ML algorithms have been developed in other branches of science for nonlinear dimensionality reduction (NDR), but have not been extensively used for fluid flows. Here, four manifold learning and two deep learning (autoencoder)-based NDR methods are investigated and compared to PCA. These are tested on two canonical fluid flow problems (laminar and turbulent) and two biomedical flows in brain aneurysms. The data reconstruction capabilities of these methods are compared, and the challenges are discussed. The temporal vs spatial arrangement of data and its influence on NDR mode extraction is investigated. Finally, the modes are qualitatively compared. The results suggest that using NDR methods would be beneficial for building more efficient reduced-order models of fluid flows. All NDR techniques resulted in smaller reconstruction errors for spatial reduction. Temporal reduction was a harder task; nevertheless, it resulted in physically interpretable modes. Our work is one of the first comprehensive comparisons of various NDR methods in unsteady flows.

The mechanics of biological fluids is an important topic in biomechanics, often requiring the use of computational tools to analyze problems with realistic geometries and material properties. This study describes the formulation and implementation of a finite element framework for computational fluid dynamics (CFD) in FEBio, a free software designed to meet the computational needs of the biomechanics and biophysics communities. This formulation models nearly incompressible flow with a compressible isothermal formulation that uses a physically realistic value for the fluid bulk modulus. It employs fluid velocity and dilatation as essential variables: The virtual work integral enforces the balance of linear momentum and the kinematic constraint between fluid velocity and dilatation, while fluid density varies with dilatation as prescribed by the axiom of mass balance. Using this approach, equal-order interpolations may be used for both essential variables over each element, contrary to traditional mixed formulations that must explicitly satisfy the inf-sup condition. The formulation accommodates Newtonian and non-Newtonian viscous responses as well as inviscid fluids. The efficiency of numerical solutions is enhanced using Broyden's quasi-Newton method. The results of finite element simulations were verified using well-documented benchmark problems as well as comparisons with other free and commercial codes. These analyses demonstrated that the novel formulation introduced in FEBio could successfully reproduce the results of other codes. The analogy between this CFD formulation and standard finite element formulations for solid mechanics makes it suitable for future extension to fluid–structure interactions (FSIs).

Vector field visualization techniques have evolved very rapidly over the last two decades, however, visualizing vector fields on complex boundary surfaces from computational flow dynamics (CFD) still remains a challenging task. In part, this is due to the large, unstructured, adaptive resolution characteristics of the meshes used in the modeling and simulation process. Out of the wide variety of existing flow field visualization techniques, vector field clustering algorithms offer the advantage of capturing a detailed picture of important areas of the domain while presenting a simplified view of areas of less importance. This paper presents a novel, robust, automatic vector field clustering algorithm that produces intuitive and insightful images of vector fields on large, unstructured, adaptive resolution boundary meshes from CFD. Our bottom-up, hierarchical approach is the first to combine the properties of the underlying vector field and mesh into a unified error-driven representation. The motivation behind the approach is the fact that CFD engineers may increase the resolution of model meshes according to importance. The algorithm has several advantages. Clusters are generated automatically, no surface parameterization is required, and large meshes are processed efficiently. The most suggestive and important information contained in the meshes and vector fields is preserved while less important areas are simplified in the visualization. Users can interactively control the level of detail by adjusting a range of clustering distance measure parameters. We describe two data structures to accelerate the clustering process. We also introduce novel visualizations of clusters inspired by statistical methods. We apply our method to a series of synthetic and complex, real-world CFD meshes to demonstrate the clustering algorithm results.

Keywords: Vector Field Visualization, Clustering, Feature-based, Surfaces

Understanding fluid flow is a difficult problem and of increasing importance as computational fluid dynamics (CFD) produces an abundance of simulation data. Experimental flow analysis has employed techniques such as shadowgraph, interferometry, and schlieren imaging for centuries, which allow empirical observation of inhomogeneous flows. Shadowgraphs provide an intuitive way of looking at small changes in flow dynamics through caustic effects while schlieren cutoffs introduce an intensity gradation for observing large scale directional changes in the flow. Interferometry tracks changes in phase-shift resulting in bands appearing. The combination of these shading effects provides an informative global analysis of overall fluid flow. Computational solutions for these methods have proven too complex until recently due to the fundamental physical interaction of light refracting through the flow field. In this paper, we introduce a novel method to simulate the refraction of light to generate synthetic shadowgraph, schlieren and interferometry images of time-varying scalar fields derived from computational fluid dynamics data. Our method computes physically accurate schlieren and shadowgraph images at interactive rates by utilizing a combination of GPGPU programming, acceleration methods, and data-dependent probabilistic schlieren cutoffs. Applications of our method to multifield data and custom application-dependent color filter creation are explored. Results comparing this method to previous schlieren approximations are finally presented.

Keywords: uintah, c-safe

In this paper, we present two combinatorial methods to process 3-D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are a straightforward extension of an existing 2-D technique to 3-D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. We evaluate our method on a CFD-simulation of a gas furnace chamber. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow.

An optimal combustion process within an engine block is central to the performance of many motorized vehicles. Associated with this process are two important patterns of flow: swirl and tumble motion, which optimize the mixing of fluid within each of an engine's cylinders. The simulation data associated with in-cylinder tumble motion within a gas engine, given on an unstructured, timevarying and adaptive resolution CFD grid, demands robust visualization methods that apply to unsteady flow. Good visualizations are necessary to analyze the simulation data of these in-cylinder flows. We present a range of methods including integral, feature-based, and image-based schemes with the goal of extracting and visualizing these two important patterns of motion. We place a strong emphasis on automatic and semi-automatic methods, including topological analysis, that require little or no user input.We make effective use of animation to visualize the time-dependent simulation data. We also describe the challenges and implementation measures necessary in order to apply the presented methods to time-varying, volumetric grids.