Q. Du
  Gateway to Qiang's research world

  Research sponsored in part by NSF-DMS, NSF-CISE, NSF-DMR, NIH-NCI, DOE-ASCR, and NSF IUCRC

Publications may be cross listed below. For more up-to-date info, check the publication list by years

Numerical solution of PDEs

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 well-posedness:
consistency, stability, convergence,
error estimates. Also interested in the
preservation of the physical, geometric
and/or topological structures/invariants.

Finite element methods Finite difference methods Finite volume methods Spectral methods

Quantized vortices - superfluidity

Recent work on Bose-Einstein Condensate - the so-called 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 Ginzburg-Landau 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 sub-atomic 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:
* to develop, refine and simplify mezoscale and macroscale models for superconductivity, via rigorous analysis, computation and asymptotic studies
* to analyze the models to gain further understanding of the properties of these models and of their solutions, and also to determine their validity and usefulness for solving physically interesting problems;
* to develop, analyze, and implement algorithms for the numerical simulation of solutions of the various models including algorithms that give gauge invariant approximations, those based on artificial bc, and those which are parallelizable
* to use our algorithms and codes to study superconducting phenomena.
(link to gallery)

Computational Materials

(research supported in part by NSF DMS and NSF ITR)
Computation of
coarsening using
G-L eq. and ETD

  Critical nuclei (link to gallery)
Phase field modeling
Computing transition states
Modeling material defect
Phase diagram
Liquid crystal
Nonlocal models

Computational biology

(research supported in part by NSF DMS)
Shape of red blood cell:
Discocytes, resulting from simulation
using phase field models
Cell membrane
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.
Cell aggregation/adhesion
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.

Computational fluid

3D Meshing and Centroidal Voronoi Tessellation

(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.

Mathematical Modelling

Structural stability
Nonlinear hyperbolic conservation laws
Stochastic and multiscale modelling

Parallel algorithms

Domain decomposition

Parallel implementaion

Divide and Conquer for eigenproblems

What's new in polynomial solving and eigen-problems?

Mathematical analysis, PDEs, Operators

 Contact Qiang Du    2003-04-08