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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.
A new closed-form solution for a radial two-layer drawdown equation for groundwater under constant-flux pumping in a finite-radius well
(2003-07-01)
Yeh, HD
;
Yang, SY
;
Peng, HY
;
環境工程研究所
;
Institute of Environmental Engineering
A mathematical model is presented for describing the groundwater flow in a radial two-layer confined aquifer system with a constant-flux pumping well that has a wellbore skin and finite well radius. The Laplace-domain solution for the model is first derived by the Laplace transforms; and the time-domain solution in terms of the aquifer drawdown is then obtained form the Laplace inversion using the Bromwich integral method. When neglecting the well radius, our Laplace-domain solution is shown to reduce to a Laplace-domain solution given by Butler [J. Hydrol. 101 (1988) 15]. A unified numerical approach including a root search approach, the Gaussian quadrature, and the Shanks method is employed for evaluating this time-domain solution. The evaluated results of the solution agree well with those of the Laplace-domain solution estimated by the modified Crump algorithm. This new solution can be used either to predict the spatial and temporal drawdown distributions in both the skin and formation zones or to investigate the effects of the skin type, skin thickness and well radius on the drawdown distribution. (C) 2003 Elsevier Science Ltd. All rights reserved.
A simple approach using Bouwer and Rice's method for slug test data analysis
(2004-09-01)
Yang, SY
;
Yeh, HD
;
環境工程研究所
;
Institute of Environmental Engineering
Slug test data obtained from tests performed in an unconfined aquifer are commonly analyzed by graphical or numerical approaches to determine the aquifer parameters. This paper derives three fourth-degree polynomials to represent the relationship between Bouwer and Rice's coefficients and the ratio of the screen length to the radius of the gravel envelope. A numerical approach using the nonlinear least squares and Newton's method is used to determine hydraulic conductivity from the best fit of the slug test data. The method of nonlinear least squares minimizes the sum of the squares of the differences between the predicted and observed water levels inside the well. With the polynomials, the hydraulic conductivity can be obtained by simply solving the nonlinear least squares equation by Newton's method. A computer code, SLUGBR, was developed from the derived polynomials using the proposed numerical approach. The results of analyzing two slug test datasets show that SLUGBR can determine hydraulic conductivity with very good accuracy.
Laplace-domain solutions for radial two-zone flow equations under the conditions of constant-head and partially penetrating well
(2005-03-01)
Yang, SY
;
Yeh, HD
;
環境工程研究所
;
Institute of Environmental Engineering
A mathematical model is presented for a constant-head test performed in a partially penetrating well with a finite-thickness skin. The model uses a no-flow boundary condition for the casing and a constant-head boundary condition for the screen to represent the partially penetrating well. The Laplace-domain solutions for the dimensionless flow rate at the wellbore and the hydraulic heads in the skin and formation zones are derived using the Laplace and finite Fourier cosine transforms. The solutions of hydraulic heads have been shown to satisfy the governing equations, related boundary conditions, and continuity requirements for the pressure head and flow rate at the interface of the skin zone and undisturbed formation. In addition, an efficient algorithm for evaluating those solutions is also presented. The dimensionless flow rates obtained from new solutions have been shown to be better than those of Novakowski's solutions, especially when the penetration ratio is large.
Phase transformations in a Cu-25at.%Al-7.5at.%Mn alloy
(2006-03-01)
Yang, SY
;
Liu, TF
;
材料科學與工程學系
;
Department of Materials Science and Engineering
The as-quenched microstructure of the Cu-25at.%Al-7.5at.%Mn alloy was D0(3) phase containing extremely fine L-J precipitates (the L-J phase is a new type of precipitate, which was firstly observed and identified by Liu and Jena (designated L-J phase) in a Cu-2.2-Mn0.8Al alloy.). When the as-quenched alloy was aged at temperatures ranging from 500 degrees C to 700 degrees C, the phase transition sequence was found to be (gamma-brass + L-J + D0(3)) (gamma-brass + L-J + B2) -> beta This result is different from that reported by previous workers in Cu-25at.%Al-(0-8)at.%Mn alloys. (c) 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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