My research interests are, Reduced Order Modeling (ROM), Scientific Computing, Lagrangian Coherent Structures (LCS), Numerical Analysis, Computational Fluid Dynamics (CFD), Numerical Methods for PDEs.

• Reduced Order Modeling (ROM)

• ROMs attempt to replace the high rank model by low dimennsional dynamcial systems, which are valuable whenever the same problem must be solved repeatedly. ROMs reduce the dimension and computational complexity of a system, using a high resolution data set obtained by either physical experiment or numerical simulation. The primary goal is to construct surrogate models that are cheap to execute and still maintain the large scale features of the original system.

My focus is ROM for nonlinear fluid flows, i.e. Navier-Stokes Equations (NSE), and turbulence.

$\begin{eqnarray} \mathbf{u}_t -Re^{-1}\Delta\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u} + \nabla p= 0 \\ \nabla\cdot \mathbf{u} = 0 \end{eqnarray}$

Standard Galerkin ROM (G-ROM) for NSE is trying to find, $\mathbf{u}_r=U+\sum_{j=1}^r{a}_j{\phi}_j$, where $\phi_j$ is the POD basis, $U$ is centering trajectory, such that

$\begin{eqnarray} (\frac{\partial\mathbf{u}_t}{\partial t},\phi_j) -\frac{2}{Re}(\frac{\nabla\mathbf{u}_r+(\nabla\mathbf{u}_r)^T}{2},\nabla\phi_j)+ ((\mathbf{u}_r\cdot\nabla)\mathbf{u}_r,\phi_j) + (\nabla p,\phi_j)= 0 \\ (\nabla\cdot \mathbf{u}_r,\phi_j) = 0 \end{eqnarray}$

G-ROM is very successful for laminar flow, but FAILED in turbulence. We designed new ROMs to address this issue, Regularization ROMs (Reg-ROM) and Approximate Deconvolution ROMs (AD-ROM).

• The Leray-ROM (LDF-ROM)
• $\begin{eqnarray} (\frac{\partial\mathbf{u}_t}{\partial t},\phi_j) -\frac{2}{Re}(\frac{\nabla\mathbf{u}_r+(\nabla\mathbf{u}_r)^T}{2},\nabla\phi_j)+ ((\overline{\mathbf{u}_r}\cdot\nabla)\mathbf{u}_r,\phi_j) + (\nabla p,\phi_j)= 0 \\ (I-\delta^2\Delta)\overline{\mathbf{u}_r=\mathbf{u}_r} \end{eqnarray}$

• The Evolve-Then-Filter ROM (EF-ROM)
• $\begin{eqnarray} (\frac{\mathbf{w}_r^{n+1}-\mathbf{u}_r^n}{\Delta t},\phi_j) -\frac{2}{Re}(\frac{\nabla\mathbf{u}_r^n+(\nabla\mathbf{u}_r^n)^T}{2},\nabla\phi_j)+ ((\mathbf{u}_r^n\cdot\nabla)\mathbf{u}_r^n,\phi_j) + (\nabla p,\phi_j)= 0 \\ (I-\delta^2\Delta)\mathbf{\overline{w}}_r^{n+1}=\mathbf{u}_r^{n+1} \end{eqnarray}$

• Approximate Deconvolution ROM (AD-ROM)
• $\begin{eqnarray} (\frac{\mathbf{w}_r^{n+1}-\mathbf{w}_r^n}{\Delta t},\phi_j) -\frac{2}{Re}(\frac{\nabla\mathbf{w}_r^n+(\nabla\mathbf{w}_r^n)^T}{2},\nabla\phi_j)+ (\overline{\mathbf{w}_r^{D,n}\mathbf{w}_r^{D,n}},\phi_j) + (\nabla p,\phi_j)= 0 \\ \mathbf{w}_r^{D,n}=\mathbf{w}_r^{AD-L,n}=(\mathbf{M}_r+\mu(\mathbf{M}_r+\delta^2\mathbf{S}_r))^{-1}(\mathbf{w}_r^n+\eta) \end{eqnarray}$

Figs. from 3D flow past a cylinder with $Re=1000$, $t=142.0s$ for different ROMs.

POD basis $\phi_1$, $\phi_2$, $\phi_3$, $\phi_6$, from left to right. For more informations, look at my publications
3D turbulent cylinder flow with $Re=1\times10^4$.
• Scientific Computing

• Quasi-Geostrophic Equations
• The Quasi-Geostrophic equations are a simple model of planet-scale fluid flow. The mathematical formula is,

$\begin{eqnarray} \frac{\partial\omega}{\partial t} + J(\omega,\psi)-\frac{1}{Ro}\frac{\partial\psi}{\partial x}=Re^{-1}\Delta\omega + F, \\ J(\omega,\psi)=\psi_y\omega_x-\psi_x\omega_y, \\ \omega=-\Delta\psi \end{eqnarray}$

The following movie is a numerical simulation of QGE with $Re=450$, $Ro=0.0036$.
• QGE-ROM
• Lagrangian Coherent Structures (LCS)

• Fluid particle trajectories are sensitive to changes in their initial conditions which makes the assessment of flow models and observations from individual tracer samples unreliable. There is a robust skeleton of material surfaces, Lagrangian Coherent Structures (LCSs), shaping those patterns. The LCS group at Caltech and references can be found here.

It starts with a velocity field $\mathbf{v}(x,t)$, $x=(x_1,x_2,x_3)\in U\subset \mathbf{R}^3$. Then the flow map is defined as,

$\begin{eqnarray} \mathbf{F}_{t_0}^t(x_0):=x(t;t_0,x_0)=\int_{t0}^t\mathbf{v}(x,t)dt \end{eqnarray}$

The Cauchy-Green train tensor is defined as,

$\begin{eqnarray} \mathbf{C}(x_0)=[\nabla\mathbf{F}_{t_0}^{t_1}(x_0)]^T\nabla\mathbf{F}_{t_0}^{t_1}(x_0) \end{eqnarray}$

The Finite-Time Lyapunov Exponent (FTLE) is computed by,

$\begin{eqnarray} \mathbf{\Lambda}_{t_0}^{t_1}(x_0)=\frac{1}{t_1-t_0}log\sqrt(\lambda_n(x_0)) \end{eqnarray}$

$\lambda_n$ is the largest eigenvalues of C-G strain tensor. There are few differences about definitions and computation in different research group, e.g. Dr. George Haller at ETH, Dr. Shawn Shadden at UC Berkeley.

Example: Double Gyre problem
$\begin{eqnarray} &\psi(x,y,t) = A\sin(\pi f(x,y))\sin(\pi y) \\ &f(x,y) = a(t)x^2+b(t)x \\ &a(t)=\epsilon\sin(\omega t)\\ &b(t)=1-2\epsilon\sin(\omega t) \end{eqnarray}$
Attracting LCSs with high resolution $1000\times500$
Repelling LCSs with high resolution $1000\times500$
• Numerical Analysis

• In part with scientific computation, I also like to prove the error estimate mathematically for the ROMs that we designed, providing a theoretical tool for people to look into.
• Computational Fluid Dynamics (CFD)

• Computational Fluid Dynamics emerge as in last 1950s, particularly in engineering, it is still at the stage of development where "problems involving complex geometries can be treated with simple physics and those involving simple geometry can be treated with complex physics" (Bailey 1986). The improvement in computer hardware, numerical algorithms as well as software has brought about the increase need to simulate more extreme physical conditions, higher Reynolds number, higher temperature in CFD, making it still one of the most popular research topics in the world. Many state-of-the-art open source numerical software and packages have been developed over the years.

Example:

Breaking of a dam 3D, simulated by OpenFOAM (if you can not see this movie, please try Safari browser)

Useful Resource
• Finite Volume Method based package OpenFOAM
• Finite Element Method based library Deal.ii
• Finite Element Method based software FEniCS
• Finite Element based PDE solver FreeFem++
• Numerical Methods for PDEs

• Finite Element method
• Finite Difference
• Spectral method