On Singularity Formation of a Nonlinear Nonlocal System (2009)
Hou, Thomas Y., Li, Congming, Shi, Zuoqiang, Wang, Shu, Yu, Xinwei
We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13]...
Presented on October 26, 2009 from 4:00 pm - 5:00 pm in Room 2 of the Paul Weber (SST) Building on the Georgia Tech campus.
We investigate the stabilizing effect of convection in three-dimensional incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations....
Thomas Y. Hou, Danping Yang, Hongyu Ran
Abstract. In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in...
In this paper, we study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the...
On the partial regularity of a 3D model of the Navier-Stokes equations (2009)
We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with...
On the stabilizing effect of convection in three-dimensional incompressible flows (2009)
We investigate the stabilizing effect of convection in three-dimensional incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations....
On the Partial Regularity of a 3D Model of the Navier-Stokes Equations (2009)
We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with...
Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data (2009)
Hou, Thomas Y., Lei, Zhen, Li, Congming
In this paper, we study the 3D axi-symmetric Navier-Stokes Equations with swirl. We prove the global regularity of the 3D Navier-Stokes equations for a family of large anisotropic initial data....
Multiscale analysis for convection dominated transport equations (2009)
In this paper, we perform a systematic multiscale analysis for convection dominated transport equations with a weak diffusion and a highly oscillatory velocity field. The paper primarily focuses on...
Stable Fourth Order Stream-Function Methods for Incompressible Flows with Boundaries (2009)
Hou, Thomas Y., Wetton, Brian R.
Fourth-order stream-function methods are proposed for the time dependent, incompressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the...
A level set formulation for the 3D incompressible Euler equations (2008)
Jian Deng, Thomas Y. Hou, Xinwei Yu
Abstract. We explore a level set representation of vorticity in the study of the singularity problems for incompressible fluid models. This representation exists for all initial vorticity fields. We...
The immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the immersed boundary method is also known to...
Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data (2008)
Hou, Thomas Y., Lei, Zhen, Li , Congming
In this paper, we study the 3D axisymmetric Navier-Stokes Equations with swirl. We prove the global regularity of the 3D Navier-Stokes equations for a family of large anisotropic initial data....
Multiscale Computations for Flow and Transport in Heterogeneous Media (2008)
Efendiev, Yalchin, Hou, Thomas Y.
Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers....
Hou, Thomas Y., Yang, Danping, Ran, Hongyu
In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in our...
A FRAMEWORK FOR MODELING SUBGRID EFFECTS FOR TWO-PHASE FLOWS IN POROUS MEDIA ∗ (2008)
Thomas Y. Hou, Andrew Westhead, Danping Yang
Abstract. In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of...
On Global Well-Posedness of the Lagrangian Averaged Euler Equations (2008)
Abstract. We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the...
Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl (2008)
In this paper, we study the dynamic stability of the 3D axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional (1D) model which approximates the...
MULTISCALE ANALYSIS AND COMPUTATION FOR THE 3D INCOMPRESSIBLE NAVIER-STOKES EQUATIONS (2008)
Thomas Y. Hou, Danping Yang, Hongyu Ran
Abstract. In this paper, we perform a systematic multiscale analysis for the 3D incompressible Navier-Stokes equations with multiscale initial data. There are two main ingredients in our multiscale...
A mathematical framework of the bridging scale method (2008)
Shaoqiang Tang, Thomas Y. Hou, Wing Kam Liu
In this paper, we present a mathematical framework of the bridging scale method (BSM), recently proposed by Liu et al. Under certain conditions, it had been designed for accurately and efficiently...
Abstract. We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global...
A level set formulation for the 3D incompressible Euler equations (2008)
Jian Deng, Thomas Y. Hou, Xinwei Yu
Abstract. We explore a level set representation of vorticity in the study of the singularity problems for incompressible fluid models. This representation exists for all initial vorticity fields. We...
The Immersed Boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to...
Abstract: We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric...
Singularity Formation in 3-D Vortex Sheets (2007)
Thomas Y. Hou, Gang Hu, Pingwen Zhang
We study singularity formation of 3-D vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D...
Thomas Y. Hou, Zhilin Li, Stanley Osher, Hongkai Zhao
In this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the...
In this article, we study existence of analytic solution for three-dimensional vortex sheets in the absence of surface tension. The key in our analysis is to derive a local leading order system,...
Organized structures, memory, and the decay of turbulence (2007)
The rapid increase in computational power has led to an unprecedented enhancement of our ability to study the behavior of complex systems in the physical, biological, and social sciences. However,...
Computing Nearly Singular Solutions Using Pseudo-Spectral Methods (2007)
In this paper, we investigate the performance of pseudo-spectral methods in computing nearly singular solutions of fluid dynamics equations. We consider two different ways of removing the aliasing...
Bridging Atomistic/Continuum Scales in Solids with Moving Dislocations (2007)
Tang, Shao-Qiang, Liu, Wing K., Karpov, Eduard G., Hou, Thomas Y.
We propose a multiscale method for simulating solids with moving dislocations. Away from atomistic subdomains where the atomistic dynamics are fully resolved, a dislocation is represented by a...
A Framework for Modeling Subgrid Effects for Two-Phase Flows in Porous Media (2006)
Hou, Thomas Y., Westhead, Andrew, Yang, Danping
In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal...
Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl (2006)
In this paper, we study the dynamic stability of the 3D axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional (1D) model which approximates the...
Numerical Study of Nearly Singular Solutions of the 3-D Incompressible Euler Equations (2006)
In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed...
On Global Well-Posedness of the Lagrangian Averaged Euler Equations (2006)
We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the bounded mean...
Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible Navier-Stokes Equations (2006)
We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the...
We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using...
Level Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic Equation (2006)
Deng, Jian, Hou, Thomas Y., Li, Ruo, Yu, Xinwei
In this article we apply the technique proposed in Deng-Hou-Yu (Comm. PDE, 2005) to study the level set dynamics of the 2D quasi-geostrophic equation. Under certain assumptions on the local geometric...
Springer Science+business Media, Thomas Y. Hou, Ruo Li
Communicated by J. E. Marsden Summary. We study the interplay between the local geometric properties and the nonblowup of the 3D incompressible Euler equations. We consider the interaction of two...
Thomas Y. Hou, Wuan Luo, Boris Rozovskii, Hao-min Zhou
In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier–Stokes equations driven by Brownian motion. The main advantage...
Global well-posedness of the viscous Boussinesq equations, Discrete Contin (2005)
(Communicated by Fang-Hua Lin) Abstract. We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in H m with m ≥ 3....
Geometric properties and nonblowup of 3D incompressible Euler flow (2005)
Jian Deng, Thomas Y. Hou, Xinwei Yu
By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum...
Thomas Y. Hou, Danping Yang, Hongyu Ran
Abstract. We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables...
Hou, Thomas Y., Wu, Xiao-Hui, Zhang, Yu
We continue the study of the nonconforming multiscale finite element method (Ms-FEM) introduced in [17, 14] for second order elliptic equations with highly oscillatory coefficients. The main...
Geometric properties and non-blowup of 3-D incompressible Euler flow (2004)
Deng, Jian, Hou, Thomas Y., Yu, Xinwei
By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum...
Thomas Y. Hou, Xiao-hui Wu, Yu Zhang
Abstract. We continue the study of the nonconforming multiscale finite element method (Ms-FEM) introduced in [17, 14] for second order elliptic equations with highly oscillatory coefficients. The...
Improved geometric conditions for nonblowup of the 3D incompressible Euler equations (2004)
Jian Deng, Thomas Y. Hou, Xinwei Yu
This is a follow-up of our recent article Deng et al. (2004). In Deng et al. (2004), we derive some local geometric conditions on vortex filaments which can prevent finite time blowup of the 3D...
Singularity formation in three-dimensional vortex sheets (2003)
Hou, Thomas Y., Hu, Gang, Zhang, Pingwen
We study singularity formation of three-dimensional (3-D) vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation...
A mixed multiscale finite element method for elliptic problems with oscillating coefficients (2003)
The recently introduced multiscale nite element method [19, 18] for solving elliptic equations with oscillating coecients is designed to capture the large scale structure of the solutions without...
A Mixed Multiscale Finite Element Method For Elliptic Problems With Oscillating Coefficients (2003)
The recently introduced multiscale nite element method [19, 18] for solving elliptic equations with oscillating coecients is designed to capture the large scale structure of the solutions without...
A mixed multiscale finite element method for elliptic problems with oscillating coefficients (2002)
The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without...
An efficient dynamically adaptive mesh for potentially singular solutions (2001)
Hector D. Ceniceros, Thomas Y. Hou
We develop an efficient dynamically adaptive mesh generator for time-dependent problems in two or more dimensions. The mesh generator is motivated by the variational approach and is based on solving...
The singular perturbation of surface tension in Hele-Shaw flows (2000)
Ceniceros, Hector D., Hou, Thomas Y.
Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidification. Very small perturbations caused by noise originate...
Convergence of a nonconforming multiscale finite element method (2000)
Efendiev, Yalchin R., Hou, Thomas Y., Wu, Xiao-Hui
The multiscale finite element method (MsFEM) [T. Y. Hou, X. H. Wu, and Z. Cai, Math. Comp., 1998, to appear; T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189] has been introduced to...
Convergence Of A Nonconforming Multiscale Finite Element Method (2000)
Yalchin R. Efendiev, Thomas Y. Hou, Xiao-Hui Wu
. The multiscale finite element method (MsFEM) [T. Y. Hou, X. H. Wu, and Z. Cai, Math. Comp., 1998, to appear; T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169--189] has been introduced...
Multiscale numerical methods for singularly-perturbed convection diffusion equations (2000)
We present an efficient and robust approach in the finite element framework for numerical solutions that exhibit multiscale behavior, with applications to singularly perturbed convection-diffusion...
Numerical study of Hele-Shaw flow with suction (1999)
Ceniceros, Hector D., Hou, Thomas Y., Si, Helen
We investigate numerically the effects of surface tension on the evolution of an initially circular blob of viscous fluid in a Hele-Shaw cell. The blob is surrounded by less viscous fluid and is...
Dynamic generation of capillary waves (1999)
Ceniceros, Hector D., Hou, Thomas Y.
We investigate the dynamic generation of capillary waves in two-dimensional, inviscid, and irrotational water waves with surface tension. It is well known that short capillary waves appear in the...
Thomas Y. Hou, Xiao-Hui Wu, Zhiqiang Cai
. We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coe#cients. The main purpose is to design a numerical method which is capable of...
The Singular Perturbation of Surface Tension in Hele-Shaw Flows (1999)
Hector D. Ceniceros, D. Cen, I Ceros, Thomas Y. Hou
nsion defines a length scale in the fingers developing in a later stage of the interface evolution. 1. Introduction Surface tension is known to play an important role in pattern formation problems...
Hou, Thomas Y., Wu, Xiao-Hui, Chen, Shiyi, Zhou, Ye
The well-known translation between the power law of the energy spectrum and that of the correlation function or the second order structure function has been widely used in analyzing random data....
Homogenization for Semilinear Hyperbolic Systems with Oscillatory Data, (1998)
The behavior of multi-dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory...
Convergence of the Point Vortex Method for the 2-D Euler Equations, (1998)
Goodman, Jonathan, Hou, Thomas Y., Lowengrub, John
We prove consistency, stability and convergence of the point vortex approximation to the 2-D incompressible Euler equations with smooth solutions. The discretization error is second-order accurate....
The objective of this study is to develop a multiscale/multiresolution method for computing wave propagation and scattering in strongly heterogeneous media. The important feature of this method is...
Multi-Scale Finite Element Approximation for Transport in Heterogeneous Porous Media (1998)
The main objective of this study is to develop an efficient multiscale coarse grid method which can be used as a competitive algorithm in studying composite materials and flow transport in strongly...
Numerical Methods for the Computation of Propagating Phase Boundaries. (1998)
Hou, Thomas Y., LeFloch, Philippe G.
The purpose of the proposed research is to develop new numerical methods for computing propagating phase boundaries in solids undergoing phase transformations, such as the austenite martensite phase...
Stability of a boundary integral method for 3-D water waves, submitted to (1998)
Abstract. We prove convergence of a modified point vortex method for timedependent water waves in a three-dimensional, inviscid, irrotational and incompressible fluid. Our stability analysis has two...
Stability of a boundary integral method for 3-D water waves, submitted to (1998)
Abstract. We prove convergence of a modified point vortex method for timedependent water waves in a three-dimensional, inviscid, irrotational and incompressible fluid. Our stability analysis has two...
H Ector, Héctor D. Ceniceros, Thomas Y. Hou
Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into...
Removing the Stiffness of Curvature in Computing 3-D Filaments (1998)
Thomas Y. Hou, Isaac Klapper, Helen Si
In this paper, we present a new formulation for computing the motion of a curvature driven 3-D filament. This new numerical method has no high order time step stability constraints that are usually...
National Aeronautics and (1998)
Space Administration Langley, Thomas Y. Hou, Thomas Y. Hou, Xiao-hui Wu, Xiao-hui Wu, Shiyi Chen, ...
The well-known translation between the power law of energy spectrum and that of the correlation function or the second order structure function has been widely used in analyzing random data. Here, we...
Stable and Efficient Numerical Approximations for Fluid Free Surfaces. (1997)
This research can be classified into two major areas. One is to develop an adaptive base finite element method for computing multiple scale solutions arising from material science and turbulent...
Thomas Hou And, Thomas Y. Hou, Xiao-hui Wu
. In this paper, we study a multiscale finite element method for solving a class of elliptic problems arising from composite materials and flows in porous media, which contain many spatial scales....
. In this paper, we study a multiscale finite element method for solving a class of elliptic problems arising from composite materials and flows in porous media, which contain many spatial scales....
A Hybrid Method For Moving Interface Problems With Application To The Hele-Shaw Flow (1997)
Thomas Y. Hou, ZHILIN LI, Stanley Osher, HONGKAI ZHAO
. In this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method [Li, 1995], is used to...
Resonance Error In Multiscale Finite Element Methods (1997)
Yalchin Efendiev, Thomas Y. Hou, Xiao-hui Wu
. A multiscale finite element method (MFEM) [9, 7] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients. This is accomplished by...
Convergence of a Boundary Integral Method for Water Waves (1996)
Beale, J. Thomas, Hou, Thomas Y., Lowengrub, John
We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without...
Professional Experience (1996)
Committee Jerrold, E. Marsden, Thomas Y. Hou, Richard M. Murray, Michael Ortiz, ...
My current research is focused on the emerging field of computational geometric mechanics, which aims to develop computational schemes that draw upon insights from geometric mechanics, through the...
Thomas Y. Hou, Xiao-Hui Wu, Zhiqiang Cai
. We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of...
A direct numerical solution of the multiple scale prob-In this paper, we study a multiscale finite element method for lems is difficult even with modern supercomputers. The solving a class of...
Russel E. Caflisch, Thomas Y. Hou, John Lowengrub
Abstract. Standard numerical methods for the Birkhoff-Rott equation for a vortex sheet are unstable due to the amplification of roundoff error by the Kelvin-Helmholtz instability. A nonlinear...
Russel E. Caflisch, Thomas Y. Hou, John Lowengrub
Standard numerical methods for the Birkhoff-Rott equation for a vortex sheet are ill-posed due to amplification of roundoff error by the Kelvin-Helmholtz instability. A nonlinear filtering method was...
Hou, Thomas Y., Wetton, Brian T. R.
A rigorous convergence result is given for a projection scheme for the Navies–Stokes equations in the presence of boundaries. The numerical scheme is based on a finite-difference approximation, and...