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“Life is good for only two things,
            discovering mathematics
                        and teaching mathematics.” --Siméon Poisson

The first time I saw the above statement was actually from a book of mathematician humor. To me, life goes on beyond mathematics. Therefore, I would not agree with Professor Poisson, other than to say that “Life is good when one is discovering and teaching mathematics.” Thus, in the process of discovery and exploration in the realm of mathematics, research and teaching will go hand in hand.

Research, to me, has become the natural companion of teaching, for I don’t teach mathematics from the textbook itself, but instead from the greater context of mathematics, from an even broader interdisciplinary perspective, or even from life. Teaching and Research are mutually enlightening. Both must be learner-centered where active involvement is imperative. Moreover, the learner must both observe and critically analyze to discover and understand.

Through this long journey, I found my joy and discovered the passion that drives me in teaching – the fuel that has now been refined in the discipline of mathematical research; the ability to ask and find my own questions and seek the solutions. I have discovered, in a deeper way than ever before, my desire to explore, to discover and to share with others. Therefore, I look forward to working as an advisor to both graduate and undergraduate students; for then, I myself can be more involved in their lives, and thus, I can involve them more in discovery, learning not just about mathematics but also becoming well-rounded people.

Research Description

               My most recent research is in the application of the recently developed canonical duality method in the context of non-convex variational problems and global optimization. These problems are directly related to a large class of semi-linear partial differential equations in mathematical physics including phase transitions, post-buckling of large deformed beam model, chaotic dynamics, nonlinear field theory, and superconductivity.

1. Multi-Scale Modeling, Finite Element Method and Global Optimization

            In a joint paper with Gao, the applicant presents a multi-scale modeling in phase transitions of solid materials with both macro and micro effects. This model is governed by a semi-linear non-convex partial differential equation. By using the canonical duality theory developed by Gao in non-convex mechanics, complicated physical phenomena were modeled and certain difficult non-convex variational problems could be converted into a coupled quadratic mixed variational problem. The extremality conditions of this variational problem are controlled by a triality theory, which reveals the multi-scale effects in phase transitions. Therefore, a potentially useful canonical dual finite element method is proposed for solving a type of non-convex variational problem. Numerical results show that while the traditional method, like the finite difference method, gives multiple equilibrium solutions – both sensitive and dependent on initial approximations. The proposed duality method provides the solution with minimum potential directly.

Figure 1: While the traditional Finite Difference Method gives 3 solutions, u_1,u_2 and u_3, the canonical dual finite element method gives solution u_min with minimum potential.

2. Non-convex Variational Analysis, Phase Transition and Numerical Computation

            Experimental observations of phase transitions and twinning in solids reveal fine layered microstructures in many configurations. In the last 30 years, considerable theoretical effort has been aimed at understanding such phenomena. This has been based mainly on finding minimizers of an energy functional that incorporates a non-convex elastic strain-energy function as the material model, similar to the van der Waals double-well potential.
            In a joint paper with Gao and Ogden, one-dimensional non-convex variational/boundary value problems are studied using a modified version of the Ericksen bar model in the context of nonlinear elastic rod theory. The strain-energy function is a general fourth-order polynomial in a suitable measure of strain that provides a convenient model for the study of, for example, phase transitions. On the basis of a canonical duality theory the nonlinear differential equation for the non-convex variational problem, here with a distributed body force with either mixed or Dirichlet boundary conditions, is converted into an algebraic equation, which can, in principle, be solved to obtain a complete set of solutions. The global minimizer and the local extrema are identified. For the soft loading device the existence of smooth solutions is discussed, and the iterative Finite Difference Method (FDM) is used to illustrate the difficulty of capturing non-smooth solutions with traditional methods. The results illustrate the important fact that smooth analytic or numerical solutions of a nonlinear mixed boundary-value problem might not be minimizers of the associated potential variational problem. From a “dual” perspective, the convergence (or non-convergence) of the FDM is explained.

Figure 2: Plots of the non-convex integrand in Ericksen's problem.

See the global minimum switching from right to left with increasing x as the transitional value x = 0.8 is traversed!

            My research (thesis) at Louisiana Tech University was primarily focused on modeling heat transfer in human skin. The applicant also has interest in statistical modeling and analysis based on his experience modeling the cyclohexene hydrogenation and dehydrogenation reactions in a continuous-flow micro reactor using regression and experimental design in SAS.

3. Heat Transfer Modeling, Finite Difference Method and Numerical Analysis

            Investigations on instantaneous skin burns are useful for an accurate assessment of burn-evaluation and for establishing thermal protections for various purposes. Meanwhile, hyperthermia with radiation is important in the treatment of cancer, and it is essential for developers and users of hyperthermia systems to predict, and interpret correctly the biomass thermal and vascular response to heating. In my joint work with Prof. Dai, we employ the well-known Pennes’ bioheat transfer equation to predict the degree of skin burn and the temperature distribution in hyperthermia cancer treatment. A fourth-order compact finite difference scheme is developed to solve Pennes' bioheat transfer equation in a three-dimensional single vessel embedded triple-layered skin structure, with two different boundary conditions (constant heating and insulation) on the top surface. To this end, we employ the fourth-order compact finite difference method to discretize the Pennes' bioheat equation, where the second-order derivatives of temperature at boundaries and interfaces are calculated using a combined compact finite difference method incorporating the boundary conditions and interfacial conditions. To demonstrate the applicability of the scheme, we investigate four physical models. Numerical results show that this compact finite difference scheme is unconditionally stable for a one-dimensional uniform-layered skin structure and more accurate than the Crank-Nicholson scheme. The comparison of the numerical results in the three-dimensional triple-layered skin structures shows that the blood vessel has a significant influence on the thermal response of the biomass. Thus, the outcomes described above provide a reliable, flexible and efficient numerical method for solving Pennes’ bioheat model in a comparatively realistic skin structure.

Figure 3: (a) Schematic configuration of a three-dimensional triple-layered skin structure and laser power. (b) Comparison of numerical errors between the second-order Crank-Nicholson scheme and the fourth-order compact finite difference scheme.