ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 21 May 2022 03:12:16 GMT2022-05-21T03:12:16Z50281- Theoretical and numerical analysis for the quasi-continuum approximation of a material particle modelhttps://scholarbank.nus.edu.sg/handle/10635/104371Title: Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model
Authors: Lin, P.
Abstract: In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.
Tue, 01 Apr 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043712003-04-01T00:00:00Z
- Spike solutions of a nonlinear electric circuit with a periodic inputhttps://scholarbank.nus.edu.sg/handle/10635/104184Title: Spike solutions of a nonlinear electric circuit with a periodic input
Authors: Chow, S.-N.; Lin, P.; Shi, S.
Abstract: We consider spike solutions of a second order differential equation with a forcing modeling a nonlinear circuit used in converting analog signals to digital ones. It is shown that the number of spikes which correspond to bits in digital signals can be provided by asymptotic expansions. Numerical results are also presented.
Thu, 01 Dec 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041842005-12-01T00:00:00Z
- Analysis of a sequential regularization method for the unsteady Navier-Stokes equationshttps://scholarbank.nus.edu.sg/handle/10635/102857Title: Analysis of a sequential regularization method for the unsteady Navier-Stokes equations
Authors: Lu, X.; Lin, P.; Liu, J.-G.
Abstract: The incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation. ©2008 American Mathematical Society.
Tue, 01 Jul 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028572008-07-01T00:00:00Z
- Sequential regularization methods for simulating mechanical systems with many closed loopshttps://scholarbank.nus.edu.sg/handle/10635/104098Title: Sequential regularization methods for simulating mechanical systems with many closed loops
Authors: Ascher, U.; Lin, P.
Abstract: The numerical simulation problem of large multibody systems has often been treated in two separate stages: (i) the forward dynamics problem for computing system accelerations from given force functions and constraints and (ii) the numerical integration problem for advancing the state in time. For the forward dynamics problem, algorithms have been given with optimal, linear complexity in the number of bodies, in case the system topology does not contain many closed loops. But the interaction between these two stages can be important. Using explicit time integration schemes, we propose a sequential regularization method (SRM) that has a linear complexity in the number of bodies per time step, even in the presence of many closed loops. The method also handles certain types of constraint singularity.
Wed, 01 Dec 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040981999-12-01T00:00:00Z
- L2-projected least-squares finite element methods for the Stokes equationshttps://scholarbank.nus.edu.sg/handle/10635/103472Title: L2-projected least-squares finite element methods for the Stokes equations
Authors: Duan, H.-Y.; Lin, P.; Saikrishnan, P.; Tan, R.C.E.
Abstract: Two new L2 least-squares (LS) finite element methods are developed for the velocity-pressure-vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition, A key feature of these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2-projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2-norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure, and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results. © 2006 Society for Industrial and Applied Mathematics.
Sun, 01 Jan 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034722006-01-01T00:00:00Z
- Simulations of singularity dynamics in liquid crystal flows: A C0 finite element approachhttps://scholarbank.nus.edu.sg/handle/10635/104114Title: Simulations of singularity dynamics in liquid crystal flows: A C0 finite element approach
Authors: Lin, P.; Liu, C.
Abstract: In this paper, we present a C0 finite element method for a 2D hydrodynamic liquid crystal model which is simpler than existing C1 element methods and mixed element formulation. The energy law is formally justified and the energy decay is used as a validation tool for our numerical computation. A splitting method combined with only a few fixed point iteration for the penalty term of the director field is applied to reduce the size of the stiffness matrix and to keep the stiffness matrix time-independent. The latter avoids solving a linear system at every time step and largely reduces the computational time, especially when direct linear system solvers are used. Our approach is verified by comparing its computational results with those obtained by C1 elements and by mixed formulation. Through numerical experiments of a few other splittings and explicit-implicit strategies, we recommend a fast and reliable algorithm for this model. A number of examples are computed to demonstrate the algorithm. © 2005 Elsevier Inc. All rights reserved.
Sat, 10 Jun 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041142006-06-10T00:00:00Z
- An operator-splitting method for a liquid crystal modelhttps://scholarbank.nus.edu.sg/handle/10635/102848Title: An operator-splitting method for a liquid crystal model
Authors: Glowinski, R.; Lin, P.; Pan, X.-B.
Abstract: In this paper an operator-splitting method is applied to find the micro-structure of a liquid crystal model with a simplified Oseen-Frank energy functional. Both projection and penalty methods are used to deal with the constant length constraint. The methods are implemented to compute director fields of liquid crystal slabs of various shapes and with various boundary data. The computational results verify researcher expectations. Some new singularity patterns are observed as well. © 2002 Elsevier Science B.V. All rights reserved.
Thu, 15 May 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028482003-05-15T00:00:00Z
- Numerical solution of quenching problems using mesh-dependent variable temporal stepshttps://scholarbank.nus.edu.sg/handle/10635/103655Title: Numerical solution of quenching problems using mesh-dependent variable temporal steps
Authors: Liang, K.W.; Lin, P.; Tan, R.C.E.
Abstract: In this paper, we introduce a new adaptive method for computing the numerical solutions of a class of quenching parabolic equations which exhibit a solution with one singularity. Our method systematically generates an irregular mesh with mesh-dependent temporal increments based on the solution behavior from which an implicit finite difference scheme associated with the irregular mesh is constructed. The convergence and stability of the finite difference scheme is analyzed for the solution before quenching. An equivalent linearized model is used to justify the stability of the method near quenching as well. A numerical example is provided to demonstrate the viability of the proposed method. © 2006 IMACS.
Tue, 01 May 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036552007-05-01T00:00:00Z
- Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking methodhttps://scholarbank.nus.edu.sg/handle/10635/103652Title: Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method
Authors: Hua, J.; Stene, J.F.; Lin, P.
Abstract: The rise of bubbles in viscous liquids is not only a very common process in many industrial applications, but also an important fundamental problem in fluid physics. An improved numerical algorithm based on the front tracking method, originally proposed by Tryggvason and his co-workers, has been validated against experiments over a wide range of intermediate Reynolds and Bond numbers using an axisymmetric model [J. Hua, J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 22 (2007) 769-795]. In the current paper, this numerical algorithm is further extended to simulate 3D bubbles rising in viscous liquids with high Reynolds and Bond numbers and with large density and viscosity ratios representative of the common air-water two-phase flow system. To facilitate the 3D front tracking simulation, mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh. On the latter mesh, the governing Navier-Stokes equations for incompressible, Newtonian flow are solved in a moving reference frame attached to the rising bubble. Specifically, the equations are solved using a finite volume scheme based on the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, and it appears to be robust even for high Reynolds numbers and high density and viscosity ratios. The 3D bubble surface is tracked explicitly using an adaptive, unstructured triangular mesh. The numerical model is integrated with the software package PARAMESH, a block-based adaptive mesh refinement (AMR) tool developed for parallel computing. PARAMESH allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer. Further, Peskin distribution function is applied to interpolate the variable values between the front and the background meshes. Detailed sensitivity analysis about the numerical modeling algorithm has been performed. The current model has also been applied to simulate a number of cases of 3D gas bubbles rising in viscous liquids, e.g. air bubbles rising in water. Simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities. In addition, we applied this model to simulate the interaction between two bubbles rising in a liquid, which illustrated the model's capability in predicting the interaction dynamics of rising bubbles. © 2007 Elsevier Inc. All rights reserved.
Sat, 01 Mar 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036522008-03-01T00:00:00Z
- Image segmentation using some piecewise constant level set methods with MBO type of projectionhttps://scholarbank.nus.edu.sg/handle/10635/103398Title: Image segmentation using some piecewise constant level set methods with MBO type of projection
Authors: Tai, X.-C.; Christiansen, O.; Lin, P.; Skjælaaen, I.
Abstract: In this work, we are trying to propose fast algorithms for Mumford-Shah image segmentation using some recently proposed piecewise constant level set methods (PCLSM). Two variants of the PCLSM will be considered in this work. The first variant, which we call the binary level set method, needs a level set function which only takes values ±1 to identify the regions. The second variant only needs to use one piecewise constant level set function to identify arbitrary number of regions. For the Mumford-Shah image segmentation model with these new level set methods, one needs to minimize some smooth energy functionals under some constrains. A penalty method will be used to deal with the constraint. AOS (additive operator splitting) and MOS (multiplicative operator splitting) schemes will be used to solve the Euler-Lagrange equations for the minimization problems. By doing this, we obtain some algorithms which are essentially applying the MBO scheme for our segmentation models. Advantages and disadvantages are discussed for the proposed schemes. © Springer Science + Business Media, LLC 2007.
Fri, 01 Jun 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033982007-06-01T00:00:00Z
- Long time numerical solution of the Navier-Stokes equations based on a sequential regularization formulationhttps://scholarbank.nus.edu.sg/handle/10635/103516Title: Long time numerical solution of the Navier-Stokes equations based on a sequential regularization formulation
Authors: Lin, P.; Liu, J.-G.; Lu, X.
Abstract: The sequential regularization method is a reformulation of the unsteady Navier-Stokes equations from the viewpoint of constrained dynamical systems or the approximate Helmholtz-Hodge projection. In this paper we study the long time behavior of the sequential regularization formulation. We give a uniform-in-time estimate between the solution of the reformulated system and that of the Navier-Stokes equations. We also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time. A couple of long time flow examples are computed to demonstrate this method. © 2008 Society for Industrial and Applied Mathematics.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035162008-01-01T00:00:00Z
- Moisture transport and diffusive instability during bread bakinghttps://scholarbank.nus.edu.sg/handle/10635/76578Title: Moisture transport and diffusive instability during bread baking
Authors: Huang, H.; Lin, P.; Zhou, W.
Abstract: In this paper we study multiphase models for simultaneous heat and mass transfer processes during bread baking. Our main objective is to provide an explanation and a remedy to the observed erroneous and/or divergent results associated with an instantaneous phase change model used in the literature. We propose a reaction-diffusion model based on the Hertz-Knudsen equation, where phase change is not instantaneous but determined by an evaporation/condensation rate. A splitting scheme is designed for the reaction-diffusion model so that a link between these two models can be established and the nonintuitive numerical instability associated with the instantaneous phase change model can be identified and eliminated. The evaporation/condensation rate is estimated by comparing results of the reaction-diffusion model with experimental observations reported in the literature. For evaporation/condensation rate beyond the estimated value, oscillatory solution with multiple regions of dry and two-phase zones is observed. We show that these are caused by an instability intrinsic to the model (which we call diffusive instability) using linear stability analysis and numerical tests. © 2007 Society for Industrial and Applied Mathematics.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/765782007-01-01T00:00:00Z
- Finite element methods based on a new formulation for the non-stationary incompressible Navier-Stokes equationshttps://scholarbank.nus.edu.sg/handle/10635/103270Title: Finite element methods based on a new formulation for the non-stationary incompressible Navier-Stokes equations
Authors: Lin, P.; Chen, X.; Ong, M.T.
Abstract: A new formulation of the Navier-Stokes equations is introduced to solve incompressible flow problems. When finite element methods are used under this formulation there is no need to worry whether Babuska-Brezzi condition is satisfied or not. Both velocity and pressure can be obtained separately and the pressure can be simply obtained by a substitution. Moreover, fully explicit time integration can be applied for easy implementation. Implementation issues are discussed and a couple of flow examples are simulated. Parallel implementation based on domain decomposition is incorporated as well. Copyright © 2004 John Wiley & Sons, Ltd.
Thu, 30 Dec 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032702004-12-30T00:00:00Z
- A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flowhttps://scholarbank.nus.edu.sg/handle/10635/102640Title: A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow
Authors: Chen, G.; Li, Z.; Lin, P.
Abstract: Biharmonic equations have many applications, especially in fluid and solid mechanics, but is difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical examples show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain. © 2007 Springer Science+Business Media, Inc.
Fri, 01 Aug 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026402008-08-01T00:00:00Z
- An adaptive homotopy multi-grid method for molecule orientations of high dimensional liquid crystalshttps://scholarbank.nus.edu.sg/handle/10635/102812Title: An adaptive homotopy multi-grid method for molecule orientations of high dimensional liquid crystals
Authors: Lin, P.; Richter, T.
Abstract: The liquid crystal molecule orientation is arranged by minimizing the so-called Oseen-Frank energy functional. For a better understanding of these complicated orientation singularities, simplified models resulting from specific choices of elastic constants are always of interest. In this paper a pseudo Newton method together with a multi-grid linear system solver or preconditioner is used to compute the orientation of liquid crystal molecules based on a simplified Oseen-Frank energy functional. The penalty method is used to deal with the unit-length constraint of liquid crystal molecules. The Newton and multi-grid methods do not converge when some parameters are small. A homotopy algorithm combined with mesh refinement strategies in order to deal with small parameter cases is studied and is found to be very robust in computing the solution of the model. The method is implemented to compute the orientation of liquid crystal molecules in domains of typical shapes and with various rotational boundary conditions in 2D and 3D. Interesting singularity patterns are observed. © 2007 Elsevier Inc. All rights reserved.
Fri, 10 Aug 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028122007-08-10T00:00:00Z
- A splitting moving mesh method for reaction-diffusion equations of quenching typehttps://scholarbank.nus.edu.sg/handle/10635/102766Title: A splitting moving mesh method for reaction-diffusion equations of quenching type
Authors: Liang, K.; Lin, P.; Ong, M.T.; Tan, R.C.E.
Abstract: This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction-diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman-Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU. © 2005 Elsevier Inc. All rights reserved.
Sat, 01 Jul 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027662006-07-01T00:00:00Z
- A predicted sequential regularization method for index-2 Hessenberg DAEshttps://scholarbank.nus.edu.sg/handle/10635/102730Title: A predicted sequential regularization method for index-2 Hessenberg DAEs
Authors: Lin, P.; Spiteri, R.J.
Abstract: The sequential regularization method (SRM) is a dynamic iterative method for the numerical solution of higher-index differential-algebraic equations (DAEs). The SRM has the advantage of being based on a regularized problem that is less stiff than those produced by standard regularization methods. Consequently, nonstiff integrators may be used, making the SRM a competitive alternative to popular integrators. In past work, the number of SRM iterations was taken to be roughly equal to the order of the numerical method used in each dynamic iteration. In this paper, we propose a predicted SRM (PSRM) that reduces the number of iterations in each dynamic iteration to one. We give a new error analysis for explicit Runge-Kutta methods applied to linear index-2 Hessenberg DAEs with or without singularities. We also give numerical examples to confirm the predicted convergence rates. For the PSRM, extrapolation formulas and methods based on the differential part of the DAEs serve as a predictor, and the SRM iteration serves as a corrector. Implementation of higher-order schemes for the PSRM makes use of continuous extensions of Runge-Kutta methods. In particular, we give a prediction scheme for the algebraic variable at intermediate stage points that suppresses order reduction in the differential variable near a singularity. Moreover, the SRM/PSRM provides new insight into operator splitting and fast convergence rates for waveform relaxation.
Tue, 01 Jan 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027302002-01-01T00:00:00Z
- Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splittinghttps://scholarbank.nus.edu.sg/handle/10635/104145Title: Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splitting
Authors: Cheng, H.; Lin, P.; Sheng, Q.; Tan, R.C.E.
Abstract: This paper studies the numerical solution of two-dimensional nonlinear degenerate reaction-diffusion differential equations with singular forcing terms over rectangular domains. The equations considered may generate strong quenching singularities. This investigation focuses on a variable time step Peaceman-Rachford splitting method for the aforementioned problem. The time adaptation is implemented based on arc-length estimations of the first time derivative of the solution. The two-dimensional problem is split into several one-dimensional problems so that the computational cost is significantly reduced. The monotonicity and localized linear stability of the variable step scheme are investigated. We give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem. It is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem.
Wed, 01 Jan 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041452003-01-01T00:00:00Z
- The local L2 projected C0 finite el ement method for Maxwell problemhttps://scholarbank.nus.edu.sg/handle/10635/104312Title: The local L2 projected C0 finite el ement method for Maxwell problem
Authors: Duan, H.-Y.; Jia, F.; Lin, P.; Tan, R.C.E.
Abstract: An element-local L2-projected C0 finite element method is presented to approximate the nonsmooth solution being not in H1 of the Maxwell problem on a nonconvex Lipschitz polyhedron with reentrant corners and edges. The key idea lies in that element-local L2 projectors are applied to both curl and div operators. The C0 linear finite element (enriched with certain higher degree bubble functions) is employed to approximate the nonsmooth solution. The coercivity in L2 norm is established uniform in the mesh-size, and the condition number O(h-2) of the resulting linear system is proven. For the solution and its curl in Hr with r < 1 we obtain an error bound O(hr) in an energy norm. Numerical experiments confirm the theoretical error bound. © 2009 Societ y for Industrial and Applied Mathematics.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043122009-01-01T00:00:00Z
- A least-squares finite element method for the magnetostatic problem in a multiply connected Lipschitz domainhttps://scholarbank.nus.edu.sg/handle/10635/102668Title: A least-squares finite element method for the magnetostatic problem in a multiply connected Lipschitz domain
Authors: Duan, H.-Y.; Lin, P.; Saikrishnan, P.; Tan, R.C.E.
Abstract: A new least-squares finite element method is developed for the curl-div magnetostatic problem in Lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials. In order to deal with possibly low regularity of the solution, local L2 projectors are introduced to standard least-squares formulation and applied to both curl and div operators. Coercivity is established by adding suitable mesh-dependent bilinear terms. As a result, any continuous finite elements (lower-order elements are enriched with suitable bubbles) can be employed. A desirable error bound is obtained: ∥u - uh∥0 ≤ C ∥u - ũ∥0, where uh and ũ are the finite element approximation and the finite element interpolant of the exact solution u, respectively. Numerical tests confirm the theoretical results. © 2007 Society for Industrial and Applied Mathematics.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026682007-01-01T00:00:00Z
- A fully explicit method for incompressible flow computationhttps://scholarbank.nus.edu.sg/handle/10635/102646Title: A fully explicit method for incompressible flow computation
Authors: Lin, P.; Guo, Q.; Chen, X.
Abstract: A new formulation of the Navier-Stokes equations is introduced to solve incompressible flow problems. It keeps the benefits of the penalty method, that is, velocity and pressure can be obtained separately and no pressure-Poisson equation is involved. Unlike the penalty method the formulation is more stable or less stiff and then explicit time integration can be applied for easy implementation. No linear or nonlinear system need be solved in the method. In the case that a large number of time steps are needed a parallelization based on domain decomposition is applied to reduce the computational time. With the explicit time integration the parallel implementation and its message passing are very simple as well. © 2003 Elsevier Science B.V. All rights reserved.
Fri, 06 Jun 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026462003-06-06T00:00:00Z
- Numerical studies of a coarse-grained approximation for dynamics of an atomic chainhttps://scholarbank.nus.edu.sg/handle/10635/103656Title: Numerical studies of a coarse-grained approximation for dynamics of an atomic chain
Authors: Lin, P.; Plecháč, P.
Abstract: In many applications, materials are modeled by a large number of particles (or atoms) where each particle interacts with all others. Near or nearest-neighbor interaction is considered to be a good simplification of the full interaction in the engineering community. However, the resulting system is still too large to be solved under the existing computer power. In this paper we shall use the finite element and/or quasicontinuum idea to both position and velocity variables in order to reduce the number of degrees of freedom. The original and approximate particle systems are related to the discretization of the virtual internal bond model (continuum model). We focus more on the discrete system since the continuum description may not be physically complete because the stress-strain relation is not monotonically increasing and thus not necessarily well posed. We provide numerical justification on how well the coarse-grained solution is close to the fine grid solution in either a viscosity-demping or a temporal-average sense. © 2007 by Begell House, Inc.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036562007-01-01T00:00:00Z
- Numerical analysis of Biot's consolidation process by radial point interpolation methodhttps://scholarbank.nus.edu.sg/handle/10635/51479Title: Numerical analysis of Biot's consolidation process by radial point interpolation method
Authors: Wang, J.G.; Liu, G.R.; Lin, P.
Abstract: An algorithm is proposed to solve Biot's consolidation problem using meshless method called a radial point interpolation method (radial PIM). The radial PIM is advantageous over the meshless methods based on moving least-square (MLS) method in implementation of essential boundary condition and over the original PIM with polynomial basis in avoiding singularity when shape functions are constructed. Two variables in Biot's consolidation theory, displacement and excess pore water pressure, are spatially approximated by the same shape functions through the radial PIM technique. Fully implicit integration scheme is proposed in time domain to avoid spurious ripple effect. Some examples with structured and unstructured nodes are studied and compared with closed-form solution or finite element method solutions. © 2002 Elsevier Science Ltd. All rights reserved.
Tue, 12 Mar 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/514792002-03-12T00:00:00Z
- Numerical solution of a virtual internal bond model for material fracturehttps://scholarbank.nus.edu.sg/handle/10635/103654Title: Numerical solution of a virtual internal bond model for material fracture
Authors: Lin, P.; Shu, C.-W.
Abstract: A virtual internal bond (VIB) model is proposed recently in mechanical engineering literatures for simulating dynamic fracture. The model is a nonlinear wave equation of mixed type (hyperbolic or elliptic). There is instability in the elliptic region and usual numerical methods might not work. We examine the artificial viscosity method for the model and apply central type schemes directly to the corresponding viscous system to ensure appropriate numerical viscous term for such a mixed type problem. We provide a formal justification of indicating convergence of the scheme despite the difficulty of the type change. The exact solution of a Riemann problem is used to demonstrate the numerical method for one-dimensional case. We then generalize the method to a two-dimensional material with a triangular or hexagonal lattice structure. Computational results for a two-dimensional example are given. © 2002 Elsevier Science B.V. All rights reserved.
Mon, 01 Jul 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036542002-07-01T00:00:00Z
- A primal-dual active-set method for non-negativity constrained total variation deblurring problemshttps://scholarbank.nus.edu.sg/handle/10635/102733Title: A primal-dual active-set method for non-negativity constrained total variation deblurring problems
Authors: Krishnan, D.; Lin, P.; Yip, A.M.
Abstract: This paper studies image deblurring problems using a total variation-based model, with a non-negativity constraint. The addition of the non-negativity constraint improves the quality of the solutions, but makes the solution process a difficult one. The contribution of our work is a fast and robust numerical algorithm to solve the non-negatively constrained problem. To overcome the nondifferentiability of the total variation norm, we formulate the constrained deblurring problem as a primal-dual program which is a variant of the formulation proposed by Chan, Golub, and Mulet for unconstrained problems. Here, dual refers to a combination of the Lagrangian and Fenchel duals. To solve the constrained primal-dual program, we use a semi-smooth Newton's method. We exploit the relationship between the semi-smooth Newton's method and the primal-dual active set method to achieve considerable simplification of the computations. The main advantages of our proposed scheme are: no parameters need significant adjustment, a standard inverse preconditioner works very well, quadratic rate of local convergence (theoretical and numerical), numerical evidence of global convergence, and high accuracy of solving the optimality system. The scheme shows robustness of performance over a wide range of parameters. A comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme. © 2007 IEEE.
Thu, 01 Nov 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027332007-11-01T00:00:00Z
- The numerical solution of a challenging class of turning point problemshttps://scholarbank.nus.edu.sg/handle/10635/104326Title: The numerical solution of a challenging class of turning point problems
Authors: Lin, P.; O'Malley Jr., R.E.
Abstract: A curious class of challenging singularly perturbed turning point problems is considered and properties of the solutions to corresponding initial value problems are studied. The solutions are exponentially small near the turning point and become unstable after passing it. Various state-of-the-art codes available in MATLAB as well as one-step and multistep methods on a uniform mesh are tested. By examining a number of examples, one finds that the usual error control strategies may not work when the solution near the turning point is small, while one-step and multistep methods on a uniform mesh work only for a moderately small perturbation parameter. A scale amplification transformation, however, seems to give the correct solution when the solution is extremely small and/or zero at the turning point. Extensions to problems with more equilibria are also briefly considered.
Sat, 01 Nov 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043262003-11-01T00:00:00Z
- Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defectshttps://scholarbank.nus.edu.sg/handle/10635/103064Title: Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects
Authors: Lin, P.
Abstract: In many applications, materials are modeled by a large number of particles (or atoms), where any particle can interact with any other. The computational cost is very high since the number of atoms is huge. Recently much attention has been paid to a so-called quasi-continuum (QC) method, which is a mixed atomistic/continuum model. The QC method uses an adaptive finite element framework to effectively integrate the majority of the atomistic degrees of freedom in regions where there is no serious defect. However, numerical analysis of this method is still in its infancy. In this paper we will conduct a convergence analysis of the QC method in the case when there is no defect. We will also remark on the case when the defect region is small. The difference between our analysis and conventional analysis is that our exact atomistic solution is not a solution of a continuous partial differential equation, but a discrete lattice scale solution which is not approximately related to any conventional partial differential equation. © 2007 Society for Industrial and Applied Mathematics.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030642007-01-01T00:00:00Z
- An energy law preserving C0 finite element scheme for simulating the kinematic effects in liquid crystal dynamicshttps://scholarbank.nus.edu.sg/handle/10635/102833Title: An energy law preserving C0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics
Authors: Lin, P.; Liu, C.; Zhang, H.
Abstract: In this paper, we use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain Ω. We reformulate the original model in the weak form which is consistent with the continuous dissipative energy law for the flow and director fields in W1, 2 + σ (Ω) (σ > 0 is an arbitrarily small number). This enables us to use convenient conformal C0 finite elements in solving the problem. Moreover, a discrete energy law is derived for a modified midpoint time discretization scheme. A fixed iterative method is used to solve the resulted nonlinear system so that a matrix free time evolution may be achieved and velocity and director variables may be solved separately. A number of hydrodynamical liquid crystal examples are computed to demonstrate the effects of the parameters and the performance of the method. © 2007 Elsevier Inc. All rights reserved.
Mon, 10 Dec 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028332007-12-10T00:00:00Z