In this proposal, the PI will develop numerical methods and their mathematical analysis, ultimately their implementations in studying wave phenomena in nano-electronics, coupled arrays of quantum dots, and phase shift masks in lithography. Propagation of classical electromagnetic and quantum waves plays a key role in these physical and engineering systems. In order to gain a quantitative understanding of the wave phenomena in those systems, accurate and efficient numerical simulations are needed with appropriately designed numerical algorithms. The targeted applications motivate our research with the following three proposed numerical methods: [1] An adaptive conservative cell average spectral method for Wigner equations in electron transport of nano-electronics; [2] A fast integral solver for quantum wave scattering in 3-D quantum dots in layered media [3] A parallel spectral element method based on eigen-oscillations for complex Helmholtz equations. The potential technology impact of this research is to understand the physics involved and provide design guidelines for nano-electronics such as nano-MOSFETs, phase shift masks, and quantum dots.
The numerical methods developed in this research will be used for the engineering design of quantum devices with significant impact on maintaining US technology preeminence in the development of new VLSI microchips, and next generation X-ray lithography in microchip manufacturing. Also, graduate students trained in this project will provide skilled workforce in the competitive high technology job market as well as potential academic researchers.
In this project, we have developed numerical algorithms on computers for describing wave propagation in inhomogeneous media, which plays a key role in various physical and engineering systems in the information age. In order to gain a quantitative understanding of the wave phenomena in those systems, accurate and efficient numerical simulations are needed with appropriately designed numerical algorithms. The outcome of this project consists of new computer algorithms for solving Maxwell equations, which govern processes in information processing devices such as microchip in personal computers, and internet and wireless communication systems such as radars and cellphones and optical fibers, and medical devices such a MRI and CAT scan. Our new numerical methods will allow engineers and scientists to simulate how waves are propagating through those systems in order to design better devices to benefit the society. Specific technical results include the following new algorithms: (a) high order hierarchical Nedelec element H(Curl) basis functions for solving time harmonic Maxwell equations; (b) well-conditioned orthogonal hierarchical L2 basis in R^n simplical elements, well-conditioned triangular and tetrahedral orthogonal basis for discontinuous Galerkin methods for time dependent PDEs; (c) H(Div) basis functions for constrained transport in magneto-hydrodynamics equations; (d) fast solver for Maxwell equations for layered media; (e ) image methods for Poisson-Boltzmann electrostatics for ion-channel; (f) a hybrid method of computing electrostatic charge over conductor (DtN mapping of Laplace equation). This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
