 
Gateway to Qiang's research world  
Research sponsored in part by NSFDMS, NSFCISE, NSFDMR, NIHNCI, DOEASCR, and NSF IUCRC 

Focusing on the design and analysis of algorithms for PDEs in various applications; Discretizations include finite element, finite difference/volume & spectral methods. Research also includes mesh generation/optimization, meshfree computation, domain decomposition and parallelization 
      
The mathematical analysis of algorithms centers on the study of wellposedness: consistency, stability, convergence, error estimates. Also interested in the preservation of the physical, geometric and/or topological structures/invariants. 
Recent work on BoseEinstein Condensate  the socalled fifth state of matter
(our computation on vortices in a
BEC are remarkably close to
experiments) 
Modeling, analysis and computations in superconductivity
Research in this direction has
focused on the study of quantized vortices in
superconductors, using mezoscale models as typified by the celebrated
GinzburgLandau equations. Though we have also worked on the microscopic
BCS models that can be used to understand the basic structure of
superconductors and of the atomic and subatomic behavior of these materials,
as well as the macroscopic vortex density and critical state models that
can be of use for the design of devices. Mezoscale models are of great
use in understanding important phenomena in superconductors such as
vortex nucleation, motion and interaction, vortex pinning,
critical fields and currents, Josephson effects, inhomogeneities and
fluctuations, vortex glass and vortex fluid structures, resistivity, etc.
Our efforts in superconductivity center on the following aspects:

Computation of coarsening using GL eq. and ETD 
Critical nuclei (link to gallery) 
Shape of red blood cell: Discocytes, resulting from simulation using phase field models 
The usual vesicle membranes are formed by
bilayers of lipid molecules.
These lipid membranes
exist everywhere in life and compartmentalize living matter into
cells and subcellular structures and present themselves as highly
structured interfaces which are essential for many biological
functions.
The equilibrium shapes of bilayer vesicle membranes have been successfully modeled via the minimization of certain shape energy such as the elastic bending energy. Recently, we have developed effective phase field bending elasticity models and simulation tools for such problems. 
We recently started collaborations with bioengineers to study the tumor metastasis. We focused on the population balance equation approach for the modeling and simulation of cell aggregation and adhesion. 
(CVT generated tetrahedral mesh for a cube with 9 balls enclosed, more pictures may be found in the papers) 
A centroidal Voronoi tessellation is a Voronoi tessellation of a given
set such that the associated generating points are centroids (centers of mass with respect to a given density function) of the corresponding Voronoi regions. Such tessellations are useful, in among many other contexts, data compression, optimal quadrature rules, optimal representation and quantization, image analysis, finite difference and volume schemes, mesh generations, optimal distribution of resources, cellular biology, and the territorial behavior of animals. We are studying methods for computing these tessellations, the underlying mathematical theory and their applications. 
Parallel implementaion