2023

K. Shukla, V. Oommen, A. Peyvan, M. Penwarden, L. Bravo, A. Ghoshal, R.M. Kirby, G. Karniadakis.
**“Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils,”** Subtitled **“arXiv:2302.00807v1,”** 2023.

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.

2022

H. Csala, S.T.M. Dawson, A. Arzani.
**“Comparing different nonlinear dimensionality reduction techniques for data-driven unsteady fluid flow modeling,”** In *Physics of Fluids*, AIP Publishing, 2022.

DOI: https://doi.org/10.1063/5.0127284

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.

2018

G.A. Ateshian, J.J. Shim, S.A. Maas, J.A. Weiss.
**“Finite Element Framework for Computational Fluid Dynamics in FEBio,”** In *Journal of Biomechanical Engineering*, Vol. 140, No. 2, ASME International, pp. 021001. Jan, 2018.

DOI: 10.1115/1.4038716

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).

2012

Z. Peng, E. Grundy, R.S. Laramee, G. Chen, N. Croft.
**“Mesh-Driven Vector Field Clustering and Visualization: An Image-Based Approach,”** In *IEEE Transactions on Visualization and Computer Graphics, 2011*, Vol. 18, No. 2, pp. 283--298. February, 2012.

DOI: 10.1109/TVCG.2011.25

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

2011

C. Brownlee, V. Pegoraro, S. Shankar, P.S. McCormick, C.D. Hansen.
**“Physically-Based Interactive Flow Visualization Based on Schlieren and Interferometry Experimental Techniques,”** In *IEEE Transactions on Visualization and Computer Graphics*, Vol. 17, No. 11, pp. 1574--1586. 2011.

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

W. Reich, Dominic Schneider, Christian Heine, Alexander Wiebel, Guoning Chen, Gerik Scheuermann.
**“Combinatorial Vector Field Topology in 3 Dimensions,”** In *Mathematical Methods in Biomedical Image Analysis (MMBIA) Proceedings IEEE MMBIA 2012*, pp. 47--59. November, 2011.

DOI: 10.1007/978-3-642-23175-9_4

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.

P.J. Smith, J. Thornock, J., D. Hinckley, M. Hradisky.
**“Large Eddy Simulation Of Industrial Flares,”** In *Proceedings of Supercomputing 2011 Companion*, pp. 137--138. 2011.

DOI: 10.1145/2148600.2148672

At the Institute for Clean and Secure Energy at the University of Utah we are focused on education through interdisciplinary research on high-temperature fuel-utilization processes for energy generation, and the associated health, environmental, policy and performance issues. We also work closely with the government agencies and private industry companies to promote rapid deployment of new technologies through the use of high performance computational tools.

Industrial flare simulation can provide important information on combustion efficiency, pollutant emissions, and operational parameter sensitivities for design or operation that cannot be measured. These simulations provide information that may help design or operate flares so as to reduce or eliminate harmful pollutants and increase combustion efficiency.

Fires and flares have been particularly difficult to simulate with traditional computational fluid dynamics (CFD) simulation tools that are based on Reynolds-Averaged Navier-Stokes (RANS) approaches. The large-scale mixing due to vortical coherent structures in these flames is not readily reduced to steady-state CFD calculations with RANS.

Simulation of combustion using Large Eddy Simulations (LES) has made it possible to more accurately simulate the complex combustion seen in these flares. Resolution of all length and time scales is not possible even for the largest supercomputers. LES gives a numerical technique which resolves the large length and time scales while using models for more homogenous smaller scales. By using LES, the combustion dynamics capture the puffing created by buoyancy in industrial flare simulation.

All of our simulations were performed using either the University of Utah's ARCHES simulation tool or the commercially available Star-CCM+ software. ARCHES is a finite-volume Large Eddy Simulation code built within the Uintah framework, which is a set of software components and libraries that facilitate the solution of partial differential equations on structured adaptive mesh refinement grids using thousands of processors. Uintah is the product of a ten-year partnership with the Department of Energy's Advanced Simulation and Computing (ASC) program through the University of Utah's Center for Simulation of Accidental Fires and Explosions (C-SAFE). The ARCHES component was initially designed for predicting the heat-flux from large buoyant pool fires with potential hazards immersed in or near a pool fire of transportation fuel. Since then, this component has been extended to solve many industrially relevant problems such as industrial flares, oxy-coal combustion processes, and fuel gasification.

In this report we showcase selected results that help us visualize and understand the physical processes occurring in the simulated systems.

Most of the simulations were completed on the University of Utah's Updraft and Ember high performance computing clusters, which are managed by the Center for High Performance Computing. High performance computational tools are essential in our effort to successfully answer all aspects of our research areas and we promote the use of high performance computational tools beyond the research environment by directly working with our industry partners.

2008

C.W. Hamman, J.C. Klewicki, R.M. Kirby.
**“On the Lamb Vector Divergence in Navier-Stokes Flows,”** In *Journal of Fluid Mechanics*, Vol. 610, pp. 261--284. 2008.

J. Krüger.
**“A GPU Framework for Interactive Simulation and Rendering of Fluid Effects,”** In *IT - Information Technology*, Vol. 4, pp. 265--268. 2008.

R.P. Pawlowski, J.P. Simonis, H.F. Walker, J.N. Shadid.
**“Inexact Newton Dogleg Methods,”** In *SIAM Journal on Numerical Analysis*, Vol. 46, No. 4, pp. 2112--2132. 2008.

DOI: 10.1137/050632166

The dogleg method is a classical trust-region technique for globalizing Newton's method. While it is widely used in optimization, including large-scale optimization via truncated-Newton approaches, its implementation in general inexact Newton methods for systems of nonlinear equations can be problematic. In this paper, we first outline a very general dogleg method suitable for the general inexact Newton context and provide a global convergence analysis for it. We then discuss certain issues that may arise with the standard dogleg implementational strategy and propose modified strategies that address them. Newton–Krylov methods have provided important motivation for this work, and we conclude with a report on numerical experiments involving a Newton–GMRES dogleg method applied to benchmark CFD problems.

2007

C. Garth, F. Gerhardt, X. Tricoche, H. Hagen.
**“Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications,”** In *Proceeding of IEEE Visualization 2007*, pp. 1464--1471. 2007.

C. Garth, B. Laramee, X. Tricoche, H. Hauser, J. Schneider.
**“Extraction and Visualization of Swirl and Tumble Motion from Engine Simulation Data,”** In * Topology-based Methods in Visualization*, Mathematics and Visualization, Springer Berlin Heidelberg, pp. 121--135. 2007.

ISBN: 978-3-540-70822-3

DOI: 10.1007/978-3-540-70823-0_9

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.

J.E. Guilkey, T.B. Harman, B. Banerjee.
**“An Eulerian-Lagrangian Approach for Simulating Explosions of Energetic Devices,”** In *Computers and Structures*, Vol. 85, No. 11-14, pp. 660--674. June-July, 2007.

DOI: 10.1016/j.compstruc.2007.01.031

An approach for the simulation of explosions of "energetic devices" is described. In this context, an energetic device is a metal container filled with a high explosive (HE). Examples include bombs, mines, rocket motors or containers used in storage and transport of HE material. Explosions may occur due to detonation or deflagration of the HE material, with initiation resulting from either mechanical or thermal input. This approach is applicable to a wide range of fluid–structure interaction scenarios, the application to energetic devices is chosen because it demonstrates the full capability of this methodology.

Simulations of this type are characterized by a number of interesting and challenging behaviors. These include the transformation of the solid HE into highly pressurized gaseous products that initially occupy regions which formerly contained only solid material. This rapid pressurization of the container leads to large deformations at high strain rates and eventual case rupture. Once the container breaks apart, the highly pressurized product gas that escapes the failing container generates shock waves that propagate through the initially quiescent surrounding fluid.

The approach, which uses a finite-volume, multi-material compressible CFD formulation, within which solid materials are represented using a particle method known as the Material Point Method, is described, including certain of the sub-grid models required to close the governing equations. Results are first presented for "rate stick" and "cylinder test" scenarios, each of which involves detonating unconfined and confined HE material, respectively. Experimental data are available for these configurations and as such they serve as validation tests. Finally, results from an unvalidated "fast cookoff" simulation in which the HE is initiated by thermal input, which causes deflagration, are shown.

C.R. Hamman, R.M. Kirby, M. Berzins.
**“Parallel Direct Simulation of Incompressible Navier Stokes Equations,”** In *Concurrency and Computation*, Vol. 19, No. 10, pp. 1403-1427. 2007.

2005

C.D. Hansen, C.R. Johnson.
**“The Visualization Handbook,”** Elsevier, 2005.

ISBN: 0-12-387582-X

C. Scheidegger, J. Comba, R. Cunha..
**“Practical CFD Simulations on the GPU Using SMAC,”** In *Computer Graphics Forum*, Vol. 24, No. 4, pp. 715--728. 2005.

2004

C. DeTar, A.L. Fogelson, C.R. Johnson, C.A. Sikorski, T. Truong.
**“Computational Engineering and Science Program at the University of Utah,”** In *Proceedings of the International Conference on Computational Science (ICCS) 2004*, Lecture Notes in Computer Science (LNCS) 3039, part 4, Edited by M. Bubak et al, pp. 1202--1209. 2004.

C. Garth, X. Tricoche, G. Scheuermann.
**“Tracking of Vector Field Singularities in Unstructured 3D Time-Dependent Datasets,”** In *Proceeding of IEEE Visualization 2004*, pp. 329--336. 2004.

C. Garth, X. Tricoche, T. Salzbrunn, T. Bobach, G. Scheuermann.
**“Surface Techniques for Vortex Visualization,”** In *Proceedings of Joint Eurographics - IEEE TCVG Symposium on Visualization*, pp. 155--164. May, 2004.

X. Tricoche, C. Garth, G. Kindlmann, E. Deines, G. Scheuermann, Markus Ruetten, Charles D. Hansen.
**“Visualization of Intricate Flow Structures for Vortex Breakdown Analysis,”** In *Proceeding of IEEE Visualization 2004*, pp. 187--194. 2004.