Research plays an important role in pushing the boundary of our knowledge of a particular subject/field. Unlike homework for your coursework, research project does not usually end when you reach a deadline, and does not have a "true" answer that is known a priori. Furthermore, since the purpose of research is to extend our knowledge, you will run into roadblocks, and the process of overcoming these roadblocks might take months or even years. As a Ph.D. student, you will spend a huge portion of your daily life working on your research project(s). So, before you decide to enroll in a doctoral program, make sure that you find your research interest and choose your doctoral research topic, research advisor, and research group wisely!!! From experience, I can tell you that working on something you love and getting along with people in your research group will make your doctoral study much more productive and enjoyable.

On this page, you can find information on my research interests and background. Details on my research lab, Scientific Computing and Fluid Dynamics Lab (SCFDL), can be found here.

Computational Fluid Dynamics (CFD)
CFD is a branch of fluid mechanics that takes advantage of computing power to analyze and study problems involving fluid flows. Strong background in mathematics, sciences, and computer programming is essential for the development and advancement of CFD. In aerospace engineering, CFD is particularly well known for its significant contribution in reducing the number of expensive wind tunnel tests required for new aircraft designs in the early 1990s. In more recent years, CFD has become one of the core tools for multidisciplinary design optimization of new, unconventional aircraft configurations. Additionally, CFD enables us to analyze flows that are hard to produce or visualize experimentally, such as analysis of atmospheric contaminant transport and visualization of turbulent flows generated by small parts of a race car. Despite decades of CFD development and its prevalent use in industry, CFD is still far from being a mature field. Indeed, more in-depth understanding and various improvements of many aspects of CFD are necessary in order for CFD to make greater impact in the overall aircraft development process.

My Research Interests
I am mainly interested in conducting research in the area of CFD and scientific computing with applications to aerospace engineering. Some research topics of interest include:
  • CFD algorithm development
  • Mesh adaptation and generation
  • Error estimation and design under uncertainty
  • High-performance computing
  • CFD software design and data analysis


Towards High-Order, Metric-Conforming Mesh Generation
Collaboration with Krzysztof Fidkwoski, Laslo Diosady, and Scott Murman

The question of whether a mesh is sufficiently fine or is of sufficient quality to achieve the desired level of accuracy and robustness is a difficult one to answer through visual inspection of the mesh, even for experienced practitioners. In fact, the heavy reliance on a user's experience in the current meshing practice has continuously caused a significant bottleneck in the CFD workflow. Taking a shortcut by simply generating a very fine mesh is also not possible since it will quickly make many problems computationally intractable. Moreover, an optimal mesh for a given discretization method is also unlikely to be optimal for a different method. Similarly, different meshes are usually needed for different output predictions or solution approximations. As computing power grows, the demand of running higher-resolution flow simulation increases, and the ability to generate higher-quality meshes becomes more important. For this project, we develop a novel meshing algorithm to generate an optimal mesh for the problem at hand in a fully automated manner. Our proposed approach is to use higher-order Riemannian metric field information as a guide to systematically determine the optimal node locations within a mesh. Unlike most meshing algorithms, this algorithm takes into account both vertices and high-order geometry nodes of every mesh element. The resulting mesh is called high-order, metric-conforming (HOMC) mesh. Preliminary results show promise for one-dimensional solution approximations and emphasize the importance of high-order geometry nodes. These findings serve as fundamental knowledge for building, a HOMC mesh generator in the future.

Improving High-Order Finite Element Approximation Through Geometrical Warping
Collaboration with Krzysztof Fidkwoski

Polynomial basis functions are the ubiquitous workhorse of high-order, finite-element methods, but their generality comes at a price of high computational cost and fragility in the face of under-resolution. Here, we present a method for constructing a posteriori tailored, generally non-polynomial, basis functions for approximating a solution and computing outputs of a system of equations. This method is similar to solution-based adaptation, in which elements of the computational mesh are sized and oriented based on characteristics of the solution. The method takes advantage of existing infrastructure in high-order methods: the reference-to-global mapping used in constructing curved elements. By optimizing this mapping, we "warp" elements to make them ideally suited for representing a target solution or computing a scalar output from the solution. Guidelines on generating a good initial guess and choosing a generalized set of optimization parameters are provided to minimize tuning time and to introduce automation into the process. For scalar advection-diffusion and Navier-Stokes problems, we show that warped elements can offer significant accuracy benefits without increasing degrees of freedom in the system.

Adjoint-Accelerated Statistical and Deterministic Inversion of Atmospheric Contaminant Transport
Collaboration with Krzysztof Fidkwoski and Ian Tobasco

In this work, we present and compare deterministic and statistical algorithms for efficiently solving large-scale contaminant source inversion problems. The underlying equations of contaminant transport are assumed linear, but unsteady, and defined over complex geometries. The algorithms presented are accelerated through discrete adjoint solutions that are pre-computed efficiently in an offline stage, yielding savings in the time-critical online stage of several orders of magnitude in computational time. In the deterministic case, adjoints accelerate the application of the Hessian matrix, while in the statistical case, adjoints are used to directly evaluate samples. To address deterioration of statistical sampling efficiency for anisotropic posteriors, we present an application of a recently developed ensemble Markov Chain Monte Carlo method. Results for two- and three-dimensional problems demonstrate the feasibility of statistical inversion for large-scale problems and show the advantage of statistical results over single-point deterministic results.