Research Center:

  • Center for Numerical Simulation & Modeling

RESEARCH AREAS

(Some of links are closed due to the space limitation)

Dr. Gao studies  modeling, methods, and theories of duality, triality,  as well as  some of closely  related concepts (such as complementarity,  polarity, symmetry and symmetry breaking, etc.) in science, engineering, and computation, with applications to general complex systems, including nonconvex/nonsmooth/discrete and nonconservative problems in database analysis, decision science, nonlinear analysis, finite deformation field theory, engineering  mechanics, global optimization and control, differential equations and geometry,  network flows and communications, energy systems, social systems,  and to large-scale and multi-scale scientific computations. His work on duality theory in convex systems emphasizes how it relates to a unified framework in natural phenomena with symmetry; while the work on triality in non-convex systems aims to understand symmetry breaking, to reveal intrinsic duality, and to discover general pattern of duality in complex systems.  His multi-disciplinary research activities were supported by Divisions of Mathematical Science (DMS), Civil and Structural Engineering (CMS), Operations Research & Production Systems (DMII), and Computer & Information Science & Engineering (CCF) at National Science Foundation. Currently he has two active NSF grants with total $210,000 for 2005-2009.  Some new grants have been approved recently for 2009-2014.
Dr. Gao was trained in the fields of Engineering Science and Applied Mathematics. His research interests range over the following areas:

Postdoctoral Research Fellows

Start Date: flexible

Description: Post-doctoral positions are available for a multidisciplinary research project (five-years) on Canonical duality theory with applications in global optimization and decision science. Candidates should have a Ph.D. in computational mathematics or related fields (operations research, applied math, computational mechanics, computer science, etc).

Please submit: CV, a research statement, and names of 3 references

 

Main Research Contributions

1.    Mathematical Modeling

·         Beam Models:

1)      elastic model (with D. Russell, click here for details),

2)      elasto-plastic model,

3)      large deformed beam models (Gao, 1996)

wtt  + k wxxxx + (ν p – a wx2 ) wxx  = f(x,t)  

This equation has at most three solutions w(x,t) at each given (x, t), represents the two possible buckling states (upper and down positions), and one unbuckled (unstable) position. These three solutions could lead to chaos in dynamical vibration for certain given data k, a, ν, the axial load p and the distributed load f(x,t). This equation is also equivalent to one-dimensional Landau-Ginzburg equation in phase transitions. Applications of this beam model in literature, please check  here (see also here)

4)       general large deformation  elastic beam models (Gao, 2000)

 

·         Multi-scale model of phase transitions in solids

·         Eigenvalue problem on optimal surface (with S.T. Yau)

·         Optimal shape design of engineering structures

·         Bi-Complementarity model and framework in general geometrical linear systems.

2.    Dynamical Systems

·         New Phenomena discovered in Chaotic Systems

a. Meta-chaos, b.  tri-chaos, and c. reason for chaos (3-D vision of chaos)

·         Understand and control chaos

·         Duality, triality, and polarity in general nonlinear dynamics

·         Canonical dual feedback control against chaos

3.    Canonical Duality Theory

This theory is composed mainly of

1)      The canonical dual transformation (nonlinear transformation), which can be used for (correctly) modeling complex systems and to formulate canonical dual problems (with zero duality gap). 

2)      The complementary-dual principle, which leads to a unified analytical solution to general problems. For detailed discussion on this principle in finite elasticity, please check  here.

3)      The triality theory, which revels an intrinsic duality pattern in complex systems, and can be used to identify both global and local extremality conditions in nonconvex variational problems and global optimization, and to predict and against chaos in nonconvex dynamical systems.

This theory is based on the original joined work with Gil Strang at MIT in 1989. It is now understood that the popular Semi-Definite Programming (SDP) method is actually a special application of this joined work. The canonical duality theory has been successfully applied to the following fields.

a.     Mathematical Analysis and Continuum Mechanics

A unified analytic solution to a class of nonconvex/boundary value problems, including

·         Large deformed elasto-plastic mechanics

·         General nonconvex/nonsmooth variational/boundary value problems (3-D)

·         Nonlinear ODEs (including Einstein’s special relativity theory)

·         Phase transitions in solids with Ray Ogden

·         Pure azimuthal shear problem with Ray Ogden

b.    Nonlinear Algebraic Systems

·         How to find all eigenvalues of a symmetric matrix 

·         Analytical solution to m-quadratic equations in n-dimensional space

·         Analytic solution to third-order nonlinear algebraic equations in n-D

c.      Global Optimization

A unified analytic solution form to a class of NP-hard problems, including

·         quadratic minimization with box/integer constraints

·         polynomial minimization 

·         0-1 programming (with S-C Fang et al)

·         fractional programming (with S.-C. Fang et al)

·         mixed integer programming (with Hanif Sherali and N. Ruan),

·         semi-infinite programming problems

·         sensor network localization

·         general nonconvex minimization problems (with H. Sherali, N. Ruan)

4.      Numerical Methods and Algorithms

1.      Pan-penalty mixed finite element method and re-scaling algorithm

2.      Canonical dual finite element method and algorithms

3.      Primal-dual algorithms for nonconvex/nonsmooth optimization problems

4.      Multi-scale and large-scale computation and algorithms.

 

 

 

Recent research results: 

·         Gao, D.Y. and Ogden, R.W. (2008) Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem, ZAMP, 59 (2008) 498–517

·         Gao, D.Y.and Ogden, R.W. (2008) Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation,  Quarterly J. Mech. Appl. Math.

·         Understand and control chaos in dynamical systems, invited lecture presented at the 9th Conf. on Dynamical Systems: Theory and Applications, Dec. 17-22, 2007, Lodz, Poland.

Editorial:

  1. Co-Editor-in-Chief for two book series:

o    Springer Book Series: 

  Advances in  Mechanics and Mathematics (AMMA)

  1. Associate Editor for the following journals

Organizations and Activity Research Groups:

Some movies about finite element simulations for dynamical post-buckling analysis of a rubber diaphragm created by my Ph.D. student Axinte Ionita. This is a 3-D nonconvex, nonsmooth, nonconservative dynamical system subjected to follower force.
 

Publications

 Complete publications list

Publications listed by research areas

Encyclopedia Articles

  1. Gao, David Y., Duality-Mathematics,   Wiley Encyclopedia of Electronical and Eletronical Engineering 6, 1999, 68-77 ps file
  2. Gao, David Y., Mono-duality in Convex Systems, ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers
  3. Gao, David Y., Bi-duality in convex dynamically systems and D.C. programming, ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers
  4. Gao, David Y., Triality in Global Optimization,  ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers

Scientific philosophy

  1. Gao, David Y.,  Dao of the complementarity-duality: I. Complementarity and dual principles in natural sciences,   in Selected Philosophical Papers of Tsinghua's Ph.D. Thesis, Tsinghua Univ. Press, 1996
  2. Gao, D.Y.,  Dao of the complementarity-duality: II.  Complementarity and dual principles in general systems.  Excellent paper award in The First National Congress on Natural Philosophy, Anhui, July 1986.   J. of Hefei University of Technology (Sociology Ed.), 2 (1986).


Grant Agents:

Links

 

Optimal Shape Design for extended beam model.
 

Back to the home of David Yang Gao

E-mail to me at gao at vt.edu