STUDY ON THREE-DIMENSIONAL FINITE BODIES CONTAINING CRACKS USING THE FINITE ELEMENT METHOD OF LINES

The three-dimensional finite element method of lines is presented, and the basic processing description of 3D FEMOL in cracking questions is given in detail. Applications to 3D bodies with cracks indicate that good accuracy can be obtained with relatively coarse girds. In particular, application to the tension specimen shows very good agreement with the evaluation of stress intensity factors, which is better than the results of other methods. This implies a considerable potential for using this method in the 3D analysis of finite geometry solids and suggests a possible extension of this technique to nonlinear material behavior.

Cracking analysis of fracture mechanics by the finite element method of lines(FEMOL)

The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor （SIF） and crack opening displacement （COD） are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems.

THE VIBRATION AND STABILITY ANALYSIS OF MODERATE THICK PLATES BY THE METHOD OF LINES

The method of lines based on Hu Hai-chang 's theory for the vibration and stability of moderate thick plates is developed. The standard nonlinear ordinary differential equation (ODE) system for natural frequencies and critical load is given by use of ODE techniques, and then any indicated eigenvalue could be obtained directly from ODE solver by employing the so-called initial eigenfunction technique instead of the mode orthogonality condition. Numerical examples show that the present method is very effective and reliable.

COMPUTATION OF STRESS INTENSITY FACTORS BY THE SUB-REGION MIXED FINITE ELEMENT METHOD OF LINES

Based on the sub-region generalized variationM principle, a sub-region mixed version of the newly-developed semi-analytical ＇finite element method of lines＇ （FEMOL） is proposed in this paper for accurate and efficient computation of stress intensity factors （SIFs） of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations （ODEs） and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.

Method of Lines for the Chiral Nonlinear Schrödinger Equation

In this paper, we solve chiral nonlinear Schrodinger equation (CNSE) numerically. Two numerical methods are derived using the explicit Runge-Kutta method of order four and the linear multistep method (Predictor-Corrector method of fourth order). The resulting schemes of fourth order accuracy in spatial and temporal directions. The CNSE is non-integrable and has two kinds of soliton solutions: bright and dark soliton. The exact solutions and the conserved quantities of CNSE are used to display the efficiency and robustness of the numerical methods we derived. Interaction of two bright solitons for different parameters is also displayed.

Heat Transfer within Direct Heating Integrated Moving Bed Torrefier by Method of Lines

This work deals with the determination of the temperature profile within a direct heating moving bed torrefier in order to determine its minimum column height. A thermal model based on eulerian-eulerian two-phase solid-gas theory was performed and solved with the method of lines. In addition, this study allows to investigate the effect of the biomass particle size on the minimum torrefier column height. This investigation was performed by changing, simultaneously, the diameter of particles and the minimum fluidization velocity of the bed. Then, the calculations were made for a counter-current torrefaction reactor of 30 cm in diameter and for 5 kg/h of the feed rate of raw sugarcane bagasse. Results show that the height of the reactor column must be at least 30 cm for that are 1 mm in diameter and 108 cm for particles that are 2 mm in diameter.

The Direct Method of Lines for Forward and Inverse Linear Elasticity Problems of Composite Materials in Star-shaped Domains

In this paper,we generalize the direct method of lines for linear elasticity problems of composite materials in star-shaped domains and consider its application to inverse elasticity problems.We assume that the boundary of the star-shaped domain can be described by an explicit C 1 parametric curve in the polar coordinate.We introduce the curvilinear coordinate,in which the irregular star-shaped domain is converted to a regular semi-infinite strip.The equations of linear elasticity are discretized with respect to the angular variable and we solve the resulting semidiscrete approximation analytically using a direct method.The eigenvalues of the semi-discrete approximation converge quickly to the true eigenvalues of the elliptic operator,which helps capture the singularities naturally.Moreover,an optimal error estimate of our method is given.For the inverse elasticity problems,we determine the Lam´e coefficients from measurement data by minimizing a regularized energy functional.We apply the direct method of lines as the forward solver in order to cope with the irregularity of the domain and possible singularities in the forward solutions.Several numerical examples are presented to show the effectiveness and accuracy of our method for both forward and inverse elasticity problems of composite materials.

Efficient numerical scheme based on the method of lines for the shallow water equations

In this paper,a nonlinear shallow-water model of tsunami wave propagation at different points along a coastline of an ocean has been numerically simulated using method of lines.The simulation is carried out for various coastal slopes and the ocean depths.The effects of the coast slope and sea depth on the tsunami wave run-up height and velocity are illustrated.The accuracy of the mathematical model is verified by solving a classical test problem with known analytic solution.The computed run-up height and velocity show satisfactory agreement with the tsunami wave physics.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time.The method presented here uses immersed finite element(IFE)functions for the discretization in spatial variables that can be carried out over a fixedmesh(such as a Cartesianmesh if desired),and this featuremakes it possible to reduce the parabolic equation to a system of ordinary differential equations(ODE)through the usual semi-discretization procedure.Therefore,with a suitable choice of the ODE solver,this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured(Cartesian)mesh.Numerical examples are presented to demonstrate features of this new method.

Simulation of electrically driven jet using Chebyshev collocation method

The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver 'ddaskr' is used to solve the ODEs and post-stabilization is executed at the end of each step.Results show the distributions of radius,linear charge density,stretching ratio and also the horizontal velocity at a time point.Meanwhile,the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.

An Efficient Approach for Simulation of Water Levels due to the Nonlinear Interaction of Tide and Surge Along the Coast of Bangladesh

The ultimate goal and highlight of this paper are to explore water levels along the coast of Bangladesh efficiently due to the nonlinear interaction of tide and surge by employing the method of lines(MOLs)with the aid of newly proposed RKAHeM(4,4)technique.In this regard,the spatial derivatives of shallow water equations(SWEs)were discretized by means of a finite difference method to obtain a system of ordinary differential equations(ODEs)of initial valued with time as an independent variable.The obtained system of ODEs was solved by the RKAHeM(4,4)technique.One-way nested grid technique was exercised to incorporate coastal complexities closely with minimum computational cost.A stable tidal oscillation was produced over the region of interest by applying the most influential tidal constituent M2 along the southern open boundary of the outer scheme.The newly developed model was applied to estimate water levels due to the non-linear interaction of tide and surge associated with the catastrophic cyclone April 1991 along the coast of Bangladesh.The approach employed in the study was found to perform well and ensure conformity with real-time observations.

An Efficient Tide-Surge Interaction Model for the Coast of Bangladesh

The numerical method of lines(MOLs)in coordination with the classical fourth-order Runge Kutta(RK(4,4))method is used to solve shallow water equations(SWEs)for foreseeing water levels owing to the nonlinear interaction of tide and surge accompanying with a storm along the coast of Bangladesh.The SWEs are developed by extending the body forces with tide generating forces(TGFs).Spatial variables of the SWEs along with the boundary conditions are approximated by means of finite difference technique on an Arakawa C-grid to attain a system of ordinary differential equations(ODEs)of initial valued in time,which are being solved with the aid of the RK(4,4)method.Nested grid technique is adopted to solve coastal complexities closely with least computational cost.A stable tidal solution in the region of our choice is produced by applying the tidal forcing with the major tidal constituent M2(lunar semi-diurnal)along the southern open-sea boundary of the outer scheme.Numerical experimentations are carried out to simulate water levels generated by the cyclonic storm AILA along the coast of Bangladesh.The model simulated results are found to be in a reasonable agreement with the limited available reported data and observations.

STRESS INTENSITY FACTORS CALCULATION IN ANTI-PLANE FRACTURE PROBLEM BY ORTHOGONAL INTEGRAL EXTRACTION METHOD BASED ON FEMOL

For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions （or between the derivatives of eigenfunctions） of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors （SIFs）, so any order SIF can be extracted based on a certain known solution of displacement （an analytic result or a numerical result）. Many numerical examples based on the finite element method of lines （FEMOL） show that the present method is very powerful and efficient.

Semi-analytic Solution of Steady Temperature Fields During Thin Plate Welding

Usually, it is very difficult to find out an analytical solution to thermal conduction problems during high temperature welding. Therefore, as an important numerical approach, the method of lines （MOLs） is introduced to solve the temperature field characterized by high gradients. The basic idea of the method is to semi-discretize the governing equation of the problem into a system of ordinary differential equations （ODEs） defined on discrete lines by means of the finite difference method, by which the thermal boundary condition with high gradients are directly embodied in formulation. Thus the temperature field can be obtained by solving the ODEs. As a numerical example, the variation of an axisymmetrical temperature field along the plate thickness can be obtained.

A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrodinger Equation

In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

A Comparative Study of Two Spatial Discretization Schemes for Advection Equation

In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studied by Von Neumann method and with the matrix analysis. The methods are applied to a number of test problems to compare the accuracy and computational efficiency. We show that both discretization techniques approximate correctly solution of advection equation and compare their accuracy and performance.

Computational Studies of Bacterial Colony Model

Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce and migrate on the dish in clumps. In this paper, we discuss a system of reaction-diffusion equations that can be used with a view to modelling this phenomenon and we solve it numerically by means of the method of lines. For the spatial discretization, we use the finite difference method and Galerkin finite element method. We present how the spatial patterns obtained depend on the spatial discretization employed and we measure the experimental order of convergence of the numerical schemes used. Further, we present the numerical results obtained by solving the model in a cubic domain.

Application of Method-of-Lines to Charging-Up Process in Pipelines with Entrapped Air

A mathematical model is presented for the charging-up process in an air-entrapped pipeline with moving boundary conditions. A coordinate transformation technique is employed to reduce fluid motion in time-dependent domains to ones in time-independent domains. The nonlinear hyperbolic partial differential equations governing the unsteady motion of fluid combined with an equation for transient shear stress between the pipe wall and the flowing fluid are solved by the method of lines. Results show that ignoring elastic effects overestimates the maximum pressure and underestimates the maximum front velocity of filling fluid. The peak pressure of the entrapped air is sensitive to the length of the initial entrapped air pocket.

Storm surge simulation along the Meghna estuarine area:an alternative approach

In this study, numerical prediction of surges associated with a storm was made through the method of lines（MOL） in coordination with the newly proposed RKARMS（4, 4） method for the meghna estuarine region, along the coast of Bangladesh. For this purpose, the vertically integrated shallow water equations（SWEs） in Cartesian coordinates were firstly transformed into ordinary differential equations（ODEs） of initial valued, which were then soloved using the new RKARMS（4, 4） method. Nested grid technique was employed for resolving the complexities of the region of interest with minimum cost. Fresh water discharge through the lower Meghna River was taken into account along the north east corner of the innermost child scheme. Numerical experiments were performed with the severe cyclone on April 1991 that crossed the coast over the study area. Simulated results by the study were found to be in good agreement with some reported data and were found to compare well with the results obtained by the MOL in addition with the classical 4th order Runge-Kutta（RK（4, 4）） method and the standard finite difference method（FDM）.

Filtered Leapfrog Time Integration with Enhanced Stability Properties

The Leapfrog method for the solution of Ordinary Differential Equation initial value problems has been historically popular for several reasons. The method has second order accuracy, requires only one function evaluation per time step, and is non-dissipative. Despite the mentioned attractive properties, the method has some unfavorable stability properties. The absolute stability region of the method is only an interval located on the imaginary axis, rather than a region in the complex plane. The method is only weakly stable and thus exhibits computational instability in long time integrations over intervals of finite length. In this work, the use of filters is examined for the purposes of both controlling the weak instability and also enlarging the size of the absolute stability region of the method.