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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.
A least-squares finite element method for incompressible flow in stress-velocity-pressure version
(1995-12-01)
Chang, CL
;
Yang, SY
;
Hsu, CH
;
交大名義發表
;
National Chiao Tung University
In this paper we are concerned with the incompressible flow in 2-D. Introducing additional variables of derivatives of velocity, which are called stresses here, the second-order dynamic equations are reduced into a first-order system with variables of stress, velocity and pressure. Combining the compatibility condition and the divergence free condition, we have a system with six first-order equations and six unknowns. Least-squares method is performed over this extended system. The analysis shows that this method achieves optimal rates of convergence in the H-1-norm as the h approaches to zero. Numerical experiences are also available.
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 two-stage least-squares finite element method for the stress-pressure-displacement elasticity equations
(1998-05-01)
Yang, SY
;
Chang, CL
;
應用數學系
;
Department of Applied Mathematics
A new stress-pressure-displacement formulation for the planar elasticity equations is proposed by introducing the auxiliary variables, stresses, and pressure. The resulting first-order system involves a nonnegative parameter that measures the material compressibility for the elastic body. A two-stage least-squares finite element procedure is introduced for approximating the solution to this system with appropriate boundary conditions. It is shown that the two-stage least-squares scheme is stable and, with respect to the order of approximation for smooth exact solutions, the rates of convergence of the approximations for all the unknowns are optimal both in the H-1-norm and in the L-2-norm. Numerical experiments with various values of the parameter are examined, which demonstrate the theoretical estimates. Among other things, computational results indicate that the behavior of convergence is uniform in the nonnegative parameter. (C) 1998 John Wiley & Sons, Inc.
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.
On the convergence and stability of the standard least squares finite element method for first-order elliptic systems
(1998-07-01)
Yang, SY
;
應用數學系
;
Department of Applied Mathematics
A general framework of the theoretical analysis for the convergence and stability of the standard least squares finite element approximations to boundary value problems of first-order linear elliptic systems is established in a natural norm. With a suitable density assumption, the standard least squares method is proved to be convergent without requiring extra smoothness of the exact solutions. The method is also shown to be stable with respect to the natural norm. Some representative problems such as the grad-div type problems and the Stokes problem are demonstrated. (C) 1998 Published by Elsevier Science Inc. All rights reserved.
Least-squares finite element approximations to the Timoshenko beam problem
(2000-10-06)
Jou, J
;
Yang, SY
;
應用數學系
;
Department of Applied Mathematics
In this paper a least-squares finite element method for the Timoshenko beam problem is proposed and analyzed. The method is shown to be convergent and stable without requiring extra smoothness of the exact solutions. For sufficiently regular exact solutions, the method achieves optimal order of convergence in the H-1-norm for all the unknowns (displacement, rotation, shear, moment), uniformly in the small parameter which is generally proportional to the ratio of thickness to length. Thus the locking phenomenon disappears as the parameter tends to zero, A sharp a posteriori error estimator which is exact in the energy norm and equivalent in the H-1-norm is also briefly discussed. (C) 2000 Published by Elsevier Science Inc. All rights reserved. AMS classification: 65N15; 65N30.
Solution for flow rates across the wellbore in a two-zone confined aquifer
(2002-02-01)
Yang, SY
;
Yeh, HD
;
環境工程研究所
;
Institute of Environmental Engineering
A closed-form solution for transient flow rates across the wellbore in a confined aquifer is derived from a two-zone radial ground-water flow equation subject to the boundary condition of keeping a constant head at the well radius. An aquifer may be considered as a two-zone system if the formation properties near the wellbore are significantly changed due to the well construction and/or well development. An efficient numerical approach is used to evaluate this newly derived solution. Values of the transient flow rate are provided in a tabular form and compared with those obtained by numerical inversion for the Laplace-domain solution. The results show that the two solutions are in good agreement. This newly derived solution can be used not only for predicting the transient flow rate across the wellbore but also for identifying the effects of a skin with a finite thickness on the estimation of transient flow rates in a ground-water system with two different formation properties.
Improved numerical evaluation of the radial groundwater flow equation
(2002-06-01)
Peng, HY
;
Yeh, HD
;
Yang, SY
;
環境工程研究所
;
Institute of Environmental Engineering
The closed-form solution for the hydraulic head, derived for the radial groundwater flow equation subject to the constant-head boundary condition at the wellbore, is in an integral form that covers an integration range from zero to infinity. The integral is difficult to evaluate due to the integrand not only consisting of the product and the square of the Bessel functions but also having a singularity at the origin. A unified numerical method is proposed to evaluate the solution with accuracy to five decimal places and for very wide ranges of dimensionless distances and times. This approach includes a singularity removal scheme, Newton's method, the Gaussian quadrature, and Shanks' method. It gives the dimensionless heads in tabular forms with better accuracy while comparing to those given by other approaches. A formula describing the flow rate across the wellbore, derived by the authors based on Darcy's law, is proved to equal those presented by Jaeger [Proc Royal Soc Edinburgh 61 (1942) 223] and Jacob and Lohman [Am Geo Union 33 (4) (1952) 559]. The same singularity removal scheme and the Gaussian quadrature are also employed to evaluate the wellbore flow rate. Computed values of dimensionless flow rate versus dimensionless time expressed in tabular forms are also correct to five decimal places. Both the tabular results of dimensionless hydraulic head and dimensionless flow rate may be useful in engineering applications. (C) 2002 Elsevier Science Ltd. All rights reserved.
Stability investigation of single-wafer process by using a spin etcher
(2003-07-01)
Kang, TK
;
Wang, CC
;
Tsui, BY
;
Yang, WL
;
Chien, FT
Etch rate instability was found in a spin etcher. When chemical was pumped to the spin etcher, the flow rate became unstable; possibly resulting in a chemical deficiency due to low mass transport at the wafer edge. Consequently, the etch rate and etch uniformity are unstable. The detailed etch mechanism has been investigated and the mass transport deficiency can be avoided. Less than 2% of etch nonuniformity and a stable process can be achieved by achieving a stable flow rate and control of the surface reaction over the whole wafer in spin etching. (C) 2003 The Electrochemical Society.

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