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Now showing 1 - 7 of 7

Object-oriented programming of adaptive finite element and finite volume methods
(1996-10-01)
Liu, JL
;
Lin, IJ
;
Shih, MZ
;
Chen, RC
;
Hsieh, MC
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.
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.
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.
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 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.
Magneto-optics of layers of semiconductor quantum dots and nano-rings
(2006-06-01)
Voskoboynikov, O
;
Wijers, CMJ
;
Liu, JL
;
Lee, CP
;
交大名義發表
We have studied the collective magneto-optical response of layers of semiconductor quantum dots and nanorings. Expressions for both the polarizability of the individual nano-objects have been used to determine the magneto-optical response on square two-dimensional lattices on those nano-objects as a function on frequency, magnetic field and angle of incidence. The magnetic field dependence of the response for layers of rings gives rise to the Aharonov-Bohm type oscillatory behavior in the reflectance and absorbance that can be measurable. Layers of dots do not display any remarkable magnetic field dependence.
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.

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