The Numerical Geometry Group
Department of Applied Mathematics and Statistics
Stony Brook University



Geometry is the foundation of many computational methods and applications across diverse spatial scales. Geometry plays a critical role in the fundamental understanding of physical principles as well as sound engineering designs. Modern applications are experiencing increasingly complex geometry, leading to significant challenges in modeling and simulation of complex physical systems. In the Numerical Geometry Group,  we strive to develop robust and novel enabling technologies for complex geometric problems in scientific and engineering applications, leveraging techniques across the traditional boundary of computer science and applied mathematics in areas including computational geometry, geometric modeling, linear algebra, numerical differential equations, data analysis, and parallel computing. The current research focus includes tracking dynamic surfaces, coupling in multi-physics analysis, computational mesh processing, high-performance geometric computing, and geometric data analysis. The application areas include solid rocket combustion, fluid-structure interactions, micro- and nano-structures, computer graphics, biomedical engineering, and medical imaging.



Affiliated Ph.D. Students

Past Students

 Tracking Dynamic Surfaces

Blasting-off rockets, flapping wings, beating hearts, floating bubbles, growing carbon nanotube... The world is full of moving objects at different temporal and spatial scales, and their modeling and simulation requires sophisticated technology to represent their dynamic shapes accurately and efficiently. Many computational problems can also be formulated and solved using moving-surface techniques. However, tracking complex moving surfaces is a very challenging problem. Popular methods include the level-set methods, which are relatively simple to implement but may suffer from insufficient accuracy for scientific applications, and front-tracking methods, which may be more accurate but are more difficult to be made robust. We are developing efficient and robust algorithms for tracking moving surfaces through a new unified theoretical framework and novel computational methods.



Past students:

Coupling in Multi-Physics Analysis

Multi-physics, multi-scale modeling is the holy grail of computational science.  Spatial and temporal coupling of multi-physics requires rigorous numerical methods supported by efficient geometric data structures and algorithms. We have developed the state-of-the-art geometric algorithms for correlating different meshes by constructing a common refinement of the meshes and advanced numerical methods for data transfer in multi-physics coupling. We have demonstrated significant advantages of our methods in fluid-structure interactions compared with the prior state of the art. We are currently further enhancing the robustness of our geometric algorithms for pathological cases and exploring efficient and stable techniques on temporal coupling.



Past students:

Computational Mesh Processing

Finite-element or finite-volume analysis requires high-quality computational meshes. Although mature commercial tools are available for generating initial  meshes, analysis and adaptation computational meshes within numerical simulation codes still poses significant challenges, especially for surface meshes with complex geometry. Our focus are primarily on feature detection and mesh optimization of surface meshes.


Past student:

High-Performance Geometric Computing

We develop geometry-aware parallel software framework and high-performance geometric and numerical algorithms. Research activities include hard-ball molecular dynamics, finite-element and boundary-element methods, parallel mesh adaptation.

Selected Publications:

Past students:

Geometric Data Analysis

Data is ubiquitous in science and engineering. The problems of interests include manifold learning, facial recognition, and extracting computational meshes from medical images.

Past Students:


Research supported by the U.S. Department of Energy through the University of California under subcontract B523819 with the University of Illinois at Urbana-Champaign (UIUC), by NSF/DARPA under CARGO grant #0310446, by Boeing through contract with UIUC, and by UIUC under subcontract C65-649 with Georgia Tech. These supports are greatly appreciated.

Last updated on 11/26/2008.