Dr. Don Liu
Louisiana Tech University, USA
Spectral Element and Mesh-free Methods: Formulations and Applications

Spectral element methods (SEM) are superior to general finite element methods (FEM) in achieving high order accuracy. Owing to orthogonal expansion and test functions in SEM, the discretization errors could be reduced exponentially to machine zeroso that SEM could achieve the spectral convergence. Depending on the specific choice of expansion and test functions, SEM could be either Nodal or Modal method. The spectral convergence property is especially desirable in achieving high (e.g. 15th order) resolution for certain applications such as resolving an electrical double layer in eletrokinetic flow or capturing asharp velocity gradient in a thin boundary layer. 

Applications of SEM are demonstrated in acquiring numerical solutions to fluid flow and heat transfer problems. Particulate flowis one form of multiphase flow occurring widely in engineering, industry, and nature. Researchers in academia and industry are interested in particulate flowsespecially for large numbers of particles, such as in sediment entrainment in coastal regions. Challenges in modeling particulate flow phenomena include tracking moving internal boundaries which are part of the solution and quantifying large amount of interactions among particles and the fluid. An economical computational physics model is developed for particulate flows with many moving particles. 

Finally, for certain problems involving large domain-deformationsuch as water-splashing and wave dam breaking, a low-order mesh-free discrete particle-based method can be effective in simulating such kind of complex problems.


Dr. Don Liu currently is the Contractor’s Trust Endowed Associate Professor in Mathematics & Statistics, and Mechanical Engineering at Louisiana Tech University, a Tier 1 national university in USA. He received his Ph.D. in Applied Mathematics (2004)from the Division of Applied Mathematics, Brown University, and his prior Ph.D. in Engineering Thermophysics (1998)from Chinese Academy of Sciences. Dr. Liu’s research has been supported by NSF with a total funding of $0.75M as PI and involved with grants of $20M. He has contributed 72publications and 13 invited talks including one keynote speech in national and international conference. 

Dr. Liu’s research areas include high order methods – spectral modal and nodal element method, medium order methods - finite element and finite difference methods, and discrete particle methods – smoothed particle hydrodynamics. In addition, Dr. Liu works on numerical solutions to partial differential equations, computational fluid dynamics, numerical heat transfer, modeling two-phase flows, computational geophysics, high performance computing with MPI, OpenMP,and parallel GPU computing.

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