Jinn-Liang Liu

Jacobi-Davidson methods for cubic eigenvalue problems

This study investigates the isothermal crystallization behaviors of polypropylene-polyethylene-(1-butene) terpolymer and the adiabatically expanded polyolefin structured foams. For this purpose, butane gas was used as a physical blowing agent. Avrami equation has been used to interpret theoretically the experimental results obtained by either DSC or polarized optical microscope. It is believed that elongation induced crystallization occurring during the adiabatic expansion process has resulted in an increase in crystallization rate, eventually leading to a faster growth rate of spherulites and an increase in the nucleation density. An analysis of the foam by SEM images showed that the structure of foam is uniform (below diameter 30 m closed cell) In addition, the thermal conductivity and the compressive strength of the polyolefin structured foams was measured. The thermal conductivity of foamed resin with excellent insulation characteristics is reduced compared with unfoamed resin. The compressive strength is decreased with increase in the expansion ratio.

Numerical methods for semiconductor heterostructures with band nonparabolicity

This article presents numerical methods for computing bound state energies and associated wave functions of three-dimensional semiconductor heterostructures with special interest in the numerical treatment of the effect of band nonparabolicity. A nonuniform finite difference method is presented to approximate a model of a cylindrical-shaped semiconductor quantum dot embedded in another semiconductor matrix. A matrix reduction method is then proposed to dramatically reduce huge eigenvalue systems to relatively very small subsystems. Moreover, the nonparabolic band structure results in a cubic type of nonlinear eigenvalue problems for which a cubic Jacobi-Davidson method with an explicit nonequivalence deflation method are proposed to compute all the desired eigenpairs. Numerical results are given to illustrate the spectrum of energy levels and the corresponding wave functions in rather detail.

Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot

Three computational techniques are presented for approximation of the ground state energy and wave function of an electron confined by a disk-shaped InAs quantum dot (QD) embedded in GaAs matrix. The problem is treated with the effective one electronic band Hamiltonian, the energy and position dependent electron effective mass approximation, and the Ben-Daniel Duke boundary conditions. To solve the three dimensional (3D) Schrodinger equation, we employ (i) the adiabatic approximation, (ii) the adiabatic approximation with averaging, and (iii) full numerical solution. It is shown that the more efficient approximations (i) and (ii) can only be used for relatively large QD sizes. The full numerical method gives qualitative as well as quantitative trends in electronic properties with various parameters.

Object-oriented programming of adaptive finite element and finite volume methods

This article describes an object-oriented implementation of the finite element method and the finite volume method in a unified adaptive system using the programming language C++. The system applies to various types of mathematical model problems. Traditionally, different numerical methods for different types of problems are implemented independently by procedural languages such as C and Fortran. Moreover, adaptive analysis programs are more complicated than nonadaptive programs. Nevertheless, these methods share many common properties such as linear system solvers, data structures, a posteriori error analyses, and refinement processes. Some advantageous features of object-oriented programming are demonstrated through the integration of these properties in the adaptive system. New data types of objects specific to adaptive methods are also introduced. The system is well-structured, extendable, and maintainable due mainly to the nature of encapsulation and inheritance of object-oriented programming.

A weighted least squares method for the backward-forward heat equation

A weighted least squares method is given for the numerical solution of parabolic partial differential equations where the diffusion coefficient changes sign. The second-order equation is transformed into a first-order system of symmetric-positive differential equations in the sense of Friedrichs and the system is solved using least squares techniques. Error estimates and some numerical examples are presented.

A new parallel adaptive finite volume method for the numerical simulation of semiconductor devices

Abstract Based on adaptive finite volume approximation, a posteriori error estimation, and monotone iteration, a novel system is proposed for parallel simulations of semiconductor devices. The system has two distinct parallel algorithms to perform a complete set of I–V simulations for any specific device model. The first algorithm is a domain decomposition on 1-irregular unstructured meshes whereas the second is a parallelization of multiple I–V points. Implemented on a Linux cluster using message passing interface libraries, both algorithms are shown to have excellent balances on dynamic loading and hence result in efficient speedup. Compared with measurement data, computational results of sub-micron MOSFET devices are given to demonstrate the accuracy and efficiency of the system.

Poisson-Nernst-Planck-Fermi theory for modeling biological ion channels

A Poisson-Nernst-Planck-Fermi (PNPF) theory is developed for studying ionic transport through biological ion channels. Our goal is to deal with the finite size of particle using a Fermi like distribution without calculating the forces between the particles, because they are both expensive and tricky to compute. We include the steric effect of ions and water molecules with nonuniform sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of water molecules in an inhomogeneous aqueous electrolyte. Including the finite volume of water and the voids between particles is an important new part of the theory presented here. Fermi like distributions of all particle species are derived from the volume exclusion of classical particles. Volume exclusion and the resulting saturation phenomena are especially important to describe the binding and permeation mechanisms of ions in a narrow channel pore. The Gibbs free energy of the Fermi distribution reduces to that of a Boltzmann distribution when these effects are not considered. The classical Gibbs entropy is extended to a new entropy form - called Gibbs-Fermi entropy - that describes mixing configurations of all finite size particles and voids in a thermodynamic system where microstates do not have equal probabilities. The PNPF model describes the dynamic flow of ions, water molecules, as well as voids with electric fields and protein charges. The model also provides a quantitative mean-field description of the charge/space competition mechanism of particles within the highly charged and crowded channel pore. The PNPF results are in good accord with experimental currents recorded in a 10(8)-fold range of Ca(2+) concentrations. The results illustrate the anomalous mole fraction effect, a signature of L-type calcium channels. Moreover, numerical results concerning water density, dielectric permittivity, void volume, and steric energy provide useful details to study a variety of physical mechanisms ranging from binding, to permeation, blocking, flexibility, and charge/space competition of the channel.

An iterative method for adaptive finite element solutions of an energy transport model of semiconductor devices

A self-adjoint formulation of the energy transport model of semiconductor devices is proposed. This new formulation leads to symmetric and monotonic properties of the resulting system of nonlinear algebraic equations from an adaptive finite element approximation of the model. A node-by-node iterative method is then presented for solving the system. This is a globally convergent method that does not require the assembly of the global matrix system and full Jacobian matrices. An adaptive algorithm implementing this method is described in detail to illustrate the main features of this paper, namely, adaptation, node-by-node calculation, and global convergence. Numerical results of simulations on deep-submicron diode and MOSFET device structures are given to demonstrate the accuracy and efficiency of the algorithm.

A Galerkin method for the forward-backward heat equation

In this paper a new variational method is proposed for the numeri- cal approximation of the solution of the forward-backward heat equation. The approach consists of first reducing the second-order problem to an equivalent first-order system, and then using a finite element procedure with continuous elements in both space and time for the numerical approximation. Under suit- able regularity assumptions, error estimates and the results of some numerical experiments are presented.

On Weak Residual Error Estimation

A general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed. Basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated. Some numerical results are given to show the effectiveness and efficiency of the estimators.