Browsing by Author "Liu, JL"

Now showing 1 - 20 of 28

A new parallel adaptive finite volume method for the numerical simulation of semiconductor devices
(2001-12-15)
Li, YM
;
Liu, JL
;
Chao, TS
;
Sze, SM
;
應用數學系
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 I-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. (C) 2001 Elsevier Science B.V. All rights reserved.
A nonconforming weak residual error estimator for elliptic partial differential equations
(1999-09-01)
Jou, J
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A nonconforming weak residual error estimator is presented and analyzed for finite element solutions of linear elliptic partial differential equations. The treatment of the flux jumps across element edges is of special interest. The estimator is obtained by solving local residual problems which do not explicitly involve the jumps and do not require boundary conditions. The estimator handles both interior and edge residuals on each element by a suitable construction of the basis functions for the local problems. Together with the previous conforming estimator, the weak residual error estimation, without the flux jumps, can thus be applied to both odd- and even-order finite element approximations.
A posteriori boundary element error estimation
(1999-06-15)
Jou, J
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
An a posteriori error estimator is presented for the boundary element method in a general framework. It is obtained by solving local residual problems for which a local concept is introduced to accommodate the fact that integral operators are nonlocal operators. The estimator is shown to have an upper and a lower bound by the constant multiples of the exact error in the energy norm for Symm's and hypersingular integral equations. Numerical results are also given to demonstrate the effectiveness of the estimator for these equations. It can be used for adaptive h, p, and hp methods. (C) 1999 Elsevier Science B.V. All rights reserved.
A posteriori finite element error analysis for symmetric positive differential equations
(1999-01-01)
Jou, J
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
Based on the solution of local weak residual problems, conforming and nonconforming error estimators are presented and analyzed for finite element solutions of symmetric positive differential equations in the sense of Friedrichs. These estimators are devised to treat the Friedrichs system in a general setting in terms of application (hyperbolic as well as mixed-type problems), approximation (h-, p and hp-version finite element methods), implementation (no local boundary conditions and no flux jumps across element boundaries) and a posteriori error analysis (very moderate conditions on the system and on the approximation). Three model problems of the Friedrichs system, namely, the neutron transport equation, the forward-backward heat equation and the Tricomi problem are used to illustrate the applicability of the weak residual error estimation.
A posteriori finite element error analysis for symmetric positive differential equations
(1999-01-01)
Jou, J
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
Based on the solution of local weak residual problems, conforming and nonconforming error estimators are presented and analyzed for finite element solutions of symmetric positive differential equations in the sense of Friedrichs. These estimators are devised to treat the Friedrichs system in a general setting in terms of application (hyperbolic as well as mixed-type problems), approximation (h-, p and hp-version finite element methods), implementation (no local boundary conditions and no flux jumps across element boundaries) and a posteriori error analysis (very moderate conditions on the system and on the approximation). Three model problems of the Friedrichs system, namely, the neutron transport equation, the forward-backward heat equation and the Tricomi problem are used to illustrate the applicability of the weak residual error estimation.
A posteriori finite element error estimators for parametrized nonlinear boundary value problems
(1996)
Liu, JL
;
Rheinboldt, WC
;
應用數學系
;
Department of Applied Mathematics
Aposteriori error estimators for finite element solutions of multi-parameter nonlinear partial differential equations are based on an element-by-element solution of local linearizations of the nonlinear equation. In general, the associated bilinear form of the linearized problems satisfies a Garding-type inequality. Under appropriate assumptions it is shown that the error estimators are bounded by constant multiples of the true error in a suitable norm. Computational experiments indicate that the estimators are effective, inexpensive, and insensitive to the choice of-the local coordinate system on the solution manifold.
A posteriori least-squares finite element error analysis for the Navier-Stokes equations
(2003-02-01)
Jou, J
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A residual type a posteriori error estimator is presented for the least-squares finite element solution of stationary incompressible Navier-Stokes equations based on the velocity-vorticity-pressure formulation with nonstandard and standard boundary conditions. Using the coerciveness of the corresponding Stokes operator and the special feature of the nonlineariry of the formulation, it is shown that,the error estimator is exact for the Stokes problem and is asymptotically exact for the Navier-Stokes problem in an energy-like norm. The resulting adaptive method is highly parallel because it does not require to assemble the global, matrix and the error estimation can be completely localized without using any information from neighboring elements.
A quantum corrected energy-transport model for nanoscale semiconductor devices
(2005-03-20)
Chen, RC
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
An energy transport model coupled with the density gradient method as quantum mechanical corrections is proposed and numerically investigated. This new model is comprehensive in both physical and mathematical aspects. It is capable of describing hot electron transport as well as significant quantum mechanical effects for advanced devices with dimensions comparable to the de Broglie wave-length. The model is completely self-adjoint for all state variables and hence provides many appealing mathematical features such as global convergence, fast iterative solution, and highly parallelizable. Numerical simulations on diode and MOSFET with the gate length down to 34 nm using this model have been performed and compared with that using the classical transport model. It is shown that the I V characteristics of this short-channel device is significantly corrected by the density-gradient equations with current drive reduced by up to 60% comparing with that of the classical model along. Moreover, a 2D quantum layer, which is only a fraction of the length scale of inversion layer, is also effectively captured by this new model with very fine mesh near the interface produced by an adaptive finite element method. (c) 2004 Elsevier Inc. All rights reserved.
A unified analysis of a weighted least squares method for first-order systems
(1998-06-01)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A unified analysis of a weighted least squares finite element method (WLSFEM) for approximating solutions of a large class of first-order differential systems is proposed. The method exhibits several advantageous features. For example, the trial and test functions are not required to satisfy the boundary conditions. Its discretization results in symmetric and positive definite algebraic systems with condition number O(h(-2) + w(2)). And a single piecewise polynomial finite element space may be used for all test and trial functions. Asymptotic convergence of the least squares approximations with suitable weights is established in a natural norm without requiring extra smoothness of the solutions. If instead, the solutions are sufficiently regular, a priori error estimates can be derived under two suitable assumptions which are related respectively to the symmetric positive systems of Friedrichs and first-order Agmon-Douglis-Nirenberg (ADN) elliptic systems. Numerous model problems fit into these two important systems. Some selective examples are examined and verified in the unified framework. (C) 1998 Published by Elsevier Science Inc. All rights reserved.
An a posteriori finite element error analysis for the Stokes equations
(2000-02-01)
Jou, J
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
Conforming and nonconforming error estimators are presented and analyzed for the mixed finite element approximation of the Stokes problem. The estimators are obtained by solving local Poisson-type problems that do not involve boundary conditions, compatibility and balancing conditions, incompressibility constraint, or flux jumps across inter-element boundaries. The estimators are bounded from above and below by constant multiples of the actual error in an energy-like norm and can be used in adaptive h, p, and hp computations. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 65N15; 65N30.
An iterative method for adaptive finite element solutions of an energy transport model of semiconductor devices
(2003-08-10)
Chen, RC
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
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. (C) 2003 Elsevier Science B.V. All rights reserved.
Analysis of least squares finite element methods for a parameter-dependent first-order system
(1998-01-01)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A parameter-dependent first-order system arising from elasticity problems is introduced. The system corresponds to the 2D isotropic elasticity equations under a stress-pressure-displacement formulation in which the nonnegative parameter measures the material compressibility for the elastic body. Standard and weighted least squares finite element methods are applied to this system, and analyses for different values of the parameter are performed in a unified manner. The methods offer certain advantages such as they need not satisfy the Babuska-Brezzi condition, a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, the resulting algebraic system is symmetric and positive definite, accurate approximations of all the unknowns can be obtained simultaneously, and, especially, computational results do not exhibit any significant numerical locking as the parameter tends to zero which corresponds to the incompressible elasticity problem (or equivalently, the Stokes problem). With suitable boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various values of the parameter are given to demonstrate the theoretical estimates.
Analysis of least squares finite element methods for a parameter-dependent first-order system
(1998-01-01)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A parameter-dependent first-order system arising from elasticity problems is introduced. The system corresponds to the 2D isotropic elasticity equations under a stress-pressure-displacement formulation in which the nonnegative parameter measures the material compressibility for the elastic body. Standard and weighted least squares finite element methods are applied to this system, and analyses for different values of the parameter are performed in a unified manner. The methods offer certain advantages such as they need not satisfy the Babuska-Brezzi condition, a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, the resulting algebraic system is symmetric and positive definite, accurate approximations of all the unknowns can be obtained simultaneously, and, especially, computational results do not exhibit any significant numerical locking as the parameter tends to zero which corresponds to the incompressible elasticity problem (or equivalently, the Stokes problem). With suitable boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various values of the parameter are given to demonstrate the theoretical estimates.
Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot
(2001-11-01)
Li, YM
;
Liu, JL
;
Voskoboynikov, O
;
Lee, CP
;
Sze, SM
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. (C) 2001 Elsevier Science B.V. All rights reserved.
Exact a posteriori error analysis of the least squares finite element method
(2000-12-01)
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A residual type a posteriori error estimator is presented for the least squares finite element method. The estimator is proved to equal the exact error in a norm induced by some least squares functional. The error indicator of each element is equal to the exact error norm restricted to the element as well. In other words, the estimator is perfectly effective and reliable for error control and for adaptive mesh refinement. The exactness property requires virtually no assumptions on the regularity of the solution and on the finite element order in the approximation or in the estimation. The least squares method is in a very general setting that applies to various linear boundary-value problems such as the elliptic systems of first-order and of even-order and the mixed type partial differential equations. Numerical results are given to demonstrate the exactness. (C) 2000 Elsevier Science Inc. All rights reserved.
Fixed-point methods for a semiconductor quantum dot model
(2004-09-01)
Hwang, TM
;
Lin, WW
;
Liu, JL
;
Wang, WC
;
應用數學系
This paper presents various fixed-point methods for computing the ground state energy and its associated wave function of a semiconductor quantum dot model. The discretization of the three-dimensional Schrodinger equation leads to a large-scale cubic matrix polynomial eigenvalue problem for which the desired eigenvalue is embedded in the interior of the spectrum. The cubic problem is reformulated in several forms so that the desired eigenpair becomes a fixed point of the new formulations. Several algorithms are then proposed for solving the fixed-point problem. Numerical results show that the simple fixed-point method with acceleration schemes can be very efficient and stable. (C) 2004 Elsevier Ltd. All rights reserved.
Interband magneto-optical transitions in a layer of semiconductor nano-rings
(2005-06-01)
Voskoboynikov, O
;
Wijers, CMJ
;
Liu, JL
;
Lee, CP
;
應用數學系
We have developed a quantitative theory of the collective electromagnetic response of layers of semiconductor nano-rings. The response can be controlled by means of an applied magnetic field through the optical Aharonov-Bohm effect and is ultimately required for the design of composite materials. We demonstrate that the local field is of paramount importance in the description of such layers. The changes in ellipsometric parameters are 2 to 3 orders of magnitude above the detection limit of a modern ellipsometer.
Jacobi-Davidson methods for cubic eigenvalue problems
(2005-09-01)
Hwang, TM
;
Lin, WW
;
Liu, JL
;
Wang, WC
;
應用數學系
Several Jacobi-Davidson type methods are proposed for computing interior eigenpairs of large-scale cubic eigenvalue problems. To successively compute the eigenpairs, a novel explicit non-equivalence deflation method with low-rank updates is developed and analysed. Various techniques such as locking, search direction transformation, restarting, and preconditioning are incorporated into the methods to improve stability and efficiency. A semiconductor quantum dot model is given as an example to illustrate the cubic nature of the eigenvalue system resulting from the finite difference approximation. Numerical results of this model are given to demonstrate the convergence and effectiveness of the methods. Comparison results are also provided to indicate advantages and disadvantages among the various methods. Copyright (c) 2004 John Wiley & Sons, Ltd.
Least-squares finite element methods for the elasticity problem
(1997-12-18)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.
Least-squares finite element methods for the elasticity problem
(1997-12-18)
Yang, SY
;
Liu, JL
;
應用數學系
;
Department of Applied Mathematics
A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H-1-norm and in the L-2-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.