Browsing by Author "Lai, MC"

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A fast iterative solver for the variable coefficient diffusion equation on a disk
(2005-09-01)
Lai, MC
;
Tseng, YH
;
應用數學系
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Department of Applied Mathematics
We present an efficient iterative method for solving the variable coefficient diffusion equation on a unit disk. The equation is written in polar coordinates and is discretized by the standard centered difference approximation under the grid arrangement by shifting half radial mesh away from the origin so that the coordinate singularity can be handled naturally without pole conditions. The resultant matrix is symmetric positive definite so the preconditioned conjugate gradient (PCG) method can be applied. Different preconditioners have been tested for comparison, in particular, a fast direct solver derived from the equation and the semi-coarsening multigrid are shown to be almost scalable with the problem size and outperform other preconditioners significantly. The present elliptic solver has been applied to study the vortex dynamics of the Ginzburg-Landau equation with a variable diffusion coefficient. (c) 2005 Elsevier Inc. All rights reserved.
A novel approach for functional coverage measurement in HDL
(2000)
Liu, CNJ
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Chang, CY
;
Jou, JY
;
Lai, MC
;
Juan, HM
While the coverage-driven functional verification is getting popular: a fast and convenient coverage measurement tool is necessary. In this paper, we propose a hovel approach for functional coverage measurement based on the VCD files produced by the simulators. The usage flow of the proposed dumpfile-based coverage analysis is much easier and smoother than that of existing instrumentation-based coverage tools. No pre-processing tool is required and no extra code will be inserted into the source code. Most importantly, the flexibility in choosing coverage metrics and measured code regions is increased. Only one simulation run is needed for any kind of coverage reports. By conducting some experiments on real examples: it shows very promising results in terms of the performance and the accuracy of coverage reports.
A simple compact fourth-order Poisson solver on polar geometry
(2002-10-10)
Lai, MC
;
應用數學系
;
Department of Applied Mathematics
We present a simple and efficient compact fourth-order Poisson solver in polar coordinates. This solver relies on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the compact fourth-order finite difference scheme. By shifting a grid a half mesh away from the origin and incorporating the symmetry constraint of Fourier coefficients, we can easily handle coordinate singularities without pole conditions. The numerical evidence confirms fourth-order accuracy for the problem on an annulus and third-order accuracy for the problem on a disk. In addition, a simple and comparably accurate approximation for the derivatives of the solution is also presented. (C) 2002 Elsevier Science (USA).
A simple Dufort-Frankel-type scheme for the Gross-Pitaevskii equation of Bose-Einstein condensates on different geometries
(2004-07-01)
Lai, MC
;
Huang, CY
;
Lin, TS
;
應用數學系
;
Department of Applied Mathematics
We develop a simple Dufort-Frankel-type scheme for solving the time-dependent Gross-Pitaevskii equation (GPE). The GPE is a nonlinear Schrodinger equation describing the Bose-Einstein condensation (BEC) at very low temperature. Three different geometries including 1D spherically symmetric, 2D cylindrically symmetric, and 3D anisotropic Cartesian domains are considered. The present finite difference method is explicit, linearly unconditional stable and is able to handle the coordinate singularities in a natural way. Furthermore, the scheme is time reversible and satisfies a discrete analogue of density conservation law. (C) 2004 Wiley Periodicals, Inc.
An integral equation method for epitaxial step-flow growth simulations
(2006-08-10)
Huang, JF
;
Lai, MC
;
Xiang, Y
;
應用數學系
;
Department of Applied Mathematics
In this paper, we describe an integral equation approach for simulating diffusion problems with moving interfaces. The solutions are represented as moving layer potentials where the unknowns are only defined on the interfaces. The resulting integro-differential equation (IDE) system is solved using spectral deferred correction (SDC) techniques developed for general differential algebraic equations (DAEs), and the time dependent potentials are evaluated efficiently using fast convolution algorithms. The numerical solver is applied to the BCF model for the epitaxial step-flow growth of crystals, for which the solutions are calculated accurately instead of using quasi-static approximations. Numerical results in 1 + 1 dimensions are compared with available results in the literature. (c) 2006 Elsevier Inc. All rights reserved.
CFMS - A collaborative file management system on WWW
(2000)
Sheu, RK
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Cheng, MC
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Chang, YS
;
Yuan, SM
;
Tsai, J
The increasing complexity and geographical separation of design data, tools and teams have demanded for a collaborative and distributed management environment. In this paper we present a practical system, the CFMS, which is designed to manage collaborative files on WWW. Wherever remote developers are, they can navigate the evolutions and relationships between target files, check in and check out them conveniently on web browsers using CFMS. The capabilities of CFMS include the two-level navigation mechanism for version selection, the relationship model that conceptually manages the relationships between physical files as well as the prefix-based naming scheme that can uniquely identify requested objects.
Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Benard convection in a box
(2003-08-01)
Chiam, KH
;
Lai, MC
;
Greenside, HS
;
應用數學系
;
Department of Applied Mathematics
An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Benard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single nonstaggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.
Fast direct solver for Poisson equation in a 2D elliptical domain
(2004-01-01)
Lai, MC
;
應用數學系
;
Department of Applied Mathematics
In this article, we extend our previous work (M.-C. Lai and W.-C. Wang, Numer Methods Partial Differential Eq 18:56-68, 2002) for developing some fast Poisson solvers on 2D polar and spherical geometries to an elliptical domain. Instead of solving the equation in an irregular Cartesian geometry, we formulate the equation in elliptical coordinates. The solver relies on representing the solution as a truncated Fourier series, then solving the differential equations of Fourier coefficients by finite difference discretizations. Using a grid by shifting half mesh away from the pole and incorporating the derived numerical boundary value, the difficulty of coordinate singularity can be elevated easily. Unlike the case of 2D disk domain, the present difference equation for each Fourier mode is coupled with its conjugate mode through the numerical boundary value near the pole; thus, those two modes are solved simultaneously. Both second- and fourth-order accurate schemes for Dirichlet and Neumann problems are presented. In particular, the fourth-order accuracy can be achieved by a three-point compact stencil which is in contrast to a five-point long stencil for the disk case. (C) 2003 Wiley Periodicals, Inc.
Fast direct solver for the biharmonic equation on a disk and its application to incompressible flows
(2005-05-25)
Lai, MC
;
Liu, HC
;
應用數學系
;
Department of Applied Mathematics
We develop a simple and efficient FFT-based fast direct solver for the biharmonic equation on a disk. The biharmonic equation is split into a coupled system of harmonic problems. We first use the truncated Fourier series expansion to derive a set of coupled singular ODEs, then we solve those singular equations by second-order finite difference discretizations. Using a radial grid with shifting a half mesh away from the origin, we can handle the coordinate singularity easily without pole conditions. The Sherman-Morrison formula is then applied to solve the resultant linear system in a cost-efficient way. The computational complexity of the method consists of O(MN log(2) N) arithmetic operations for M x N grid points. The numerical accuracy check and some applications to the incompressible Navier-Stokes flows inside a disk are conducted. (c) 2004 Elsevier Inc. All rights reserved.
Fast Poisson solver in a three-dimensional ellipsoid
(2003)
Lai, MC
;
交大名義發表
;
National Chiao Tung University
We present a simple and efficient fast direct solver for 3D Poisson problem in elliptical coordinates. The method relies on representing the solution as a truncated Fourier series, then solving the partial differential equation of Fourier coefficients by second-order finite difference discretizations. Using a grid by shifting half mesh away from the poles and incorporating the derived numerical boundary values, the difficulty of coordinate singularities can be elevated easily. Besides, the resulting linear equations can be solved very efficiently by some existing numerical algorithms.
Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain
(2006-06-15)
Lai, MC
;
Li, ZL
;
Lin, XB
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應用數學系
;
Department of Applied Mathematics
In this paper, numerical methods are proposed for Poisson equations defined in a finite or infinite domain in three dimensions. In the domain, there can exists an interface across which the source term, the flux, and therefore the solution may be discontinuous. The existence and uniqueness of the solution are also discussed. To deal with the discontinuity in the source term and in the flux, the original problem is transformed to a new one with a smooth solution. Such a transformation can be carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three-dimensional function. An auxiliary sphere is used to separate the infinite region into an interior and exterior domain. The Kelvin's inversion is used to map the exterior domain into an interior domain. The two Poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously. By choosing the mesh size carefully and exploiting the fast Fourier transform, the resulting finite difference equations can be solved efficiently. The approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain. Numerical results demonstrate second order accuracy of our algorithms. (c) 2005 Elsevier B.V. All rights reserved.
Fourth-order finite difference scheme for the incompressible Navier-Stokes equations in a disk
(2003-07-20)
Lai, MC
;
應用數學系
;
Department of Applied Mathematics
We develop an efficient fourth-order finite difference method for solving the incompressible Navier-Stokes equations in the vorticity-stream function formulation on a disk. We use the fourth-order Runge-Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth-order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented. Copyright (C) 2003 John Wiley Sons, Ltd.
New formulations for interface problems in polar coordinates
(2003)
Li, ZL
;
Wang, WC
;
Chern, IL
;
Lai, MC
;
應用數學系
In this paper, numerical methods are proposed for some interface problems in polar or Cartesian coordinates. The new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution into a problem with a smooth solution. The new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution. In conjunction with the fast immersed interface method, a fast solver has been developed for the interface problems with a piecewise constant but a discontinuous coefficient using the new formulation in a polar coordinate system.
Synchronous navigation control for distance learning on the Web
(1996-05-01)
Yeh, PJ
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Chen, BH
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Lai, MC
;
Yuan, SM
;
交大名義發表
Most of the research on ''distance learning on the Web'' is learner-centric. That is, learners play an active role in their learning process, while instructors are passive oi even invisible. Unable to treat the instructor/learner relationship as a whole causes some problems. Therefore, we introduce a new concept of instructor-oriented navigation control on the Web, and then propose and implement a protocol for it. In this model, an instructor (technically, master) can setup a learning group, and those who wish to be guided distantly (technically, slaves) during the study may have their navigation activities controlled synchronously by joining the group. We believe that instructors will become more active and positive with the help of our proposed model in a WWW-based educational environment.
Synthesis of novel triacetal trioxa-cage compounds by ozonolysis of bicyclo[2.2.1]heptenes and bicyclo[2.2.2]octenes
(1996-04-01)
Wu, CY
;
Lin, CC
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Lai, MC
;
Wu, HJ
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交大名義發表
Several novel triacetal trioxa-cage compounds 11a-11g, 13, 15, 17 and 19 are synthesized in a short sequence. Ozonolysis of bicyclo[2.2.1]heptenes 4a-4c, 8a-8c, 10a, 10b and 14 in dichloromethane at -78 degrees C followed by reduction with dimethyl sulfide gave triacetal era-cages 11a-11g, 13 and 15 in moderate yields, respectively. Ozonolysis of bicycle[2.2.2] octenes 16 and 18 under the same reaction conditions gave triacetal era-cages 17 and 19 in moderate yields, respectively.