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Initial Value Problem (initial + value_problem)
Selected AbstractsDamage-viscoplastic consistency model for rock fracture in heterogeneous rocks under dynamic loadingINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 10 2010Timo Saksala Abstract This paper presents a damage-viscoplastic consistency model for numerical simulation of brittle fracture in heterogeneous rocks. The model is based on a combination of the recent viscoplastic consistency model by Wang and the isotropic damage concept with separate damage variables in tension and compression. This approach does not suffer from ill-posedness, caused by strain softening, of the underlying boundary/initial value problem since viscoplasticity provides the regularization by introducing a length scale effect under dynamic loading conditions. The model uses the Mohr,Coulomb yield criterion with the Rankine criterion as a tensile cut-off. The damage law in compression is calibrated via the degradation index concept of Fang and Harrison. Thereby, the model is able to capture the brittle-to-ductile transition occurring in confined compression at a certain level of confinement. The heterogeneity of rock is accounted for by the statistical approach based on the Weibull distribution. Numerical simulations of confined compression test in plane strain conditions demonstrate a good agreement with the experiments at both the material point and structural levels as the fracture modes are realistically predicted. Copyright © 2009 John Wiley & Sons, Ltd. [source] Boundary Perturbation Methods for Water WavesGAMM - MITTEILUNGEN, Issue 1 2007David P. Nicholls Abstract The most successful equations for the modeling of ocean wave phenomena are the free,surface Euler equations. Their solutions accurately approximate a wide range of physical problems from open,ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, "Boundary Perturbation" methods for the analysis and numerical simulation of this "water wave problem". Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Slope stability analysis based on elasto-plastic finite element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2005H. Zheng Abstract The paper deals with two essential and related closely processes involved in the finite element slope stability analysis in two-dimensional problems, i.e. computation of the factors of safety (FOS) and location of the critical slide surfaces. A so-called ,,v inequality, sin ,,1 , 2v is proved for any elasto-plastic material satisfying Mohr,Coulomb's yield criterion. In order to obtain an FOS of high precision with less calculation and a proper distribution of plastic zones in the critical equilibrium state, it is stated that the Poisson's ratio v should be adjusted according to the principle that the ,,v inequality always holds as reducing the strength parameters c and ,. While locating the critical slide surface represented by the critical slide line (CSL) under the plane strain condition, an initial value problem of a system of ordinary differential equations defining the CSL is formulated. A robust numerical solution for the initial value problem based on the predictor,corrector method is given in conjunction with the necessary and sufficient condition ensuring the convergence of solution. A simple example, the kinematic solution of which is available, and a challenging example from a hydraulic project in construction are analysed to demonstrate the effectiveness of the proposed procedures. Copyright © 2005 John Wiley & Sons, Ltd. [source] Using a piecewise linear bottom to fit the bed variation in a laterally averaged, z -co-ordinate hydrodynamic modelINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2004XinJian Chen Abstract In developing a 3D or laterally averaged 2D model for free-surface flows using the finite difference method, the water depth is generally discretized either with the z -co-ordinate (z -levels) or a transformed co-ordinate (e.g. the so-called , -co-ordinate or , -levels). In a z -level model, the water depth is discretized without any transformation, while in a , -level model, the water depth is discretized after a so-called , -transformation that converts the water column to a unit, so that the free surface will be 0 (or 1) and the bottom will be -1 (or 0) in the stretched co-ordinate system. Both discretization methods have their own advantages and drawbacks. It is generally not conclusive that one discretization method always works better than the other. The biggest problem for the z -level model normally stems from the fact that it cannot fit the topography properly, while a , -level model does not have this kind of a topography-fitting problem. To solve the topography-fitting problem in a laterally averaged, 2D model using z -levels, a piecewise linear bottom is proposed in this paper. Since the resulting computational cells are not necessarily rectangular looking at the x,z plane, flux-based finite difference equations are used in the model to solve the governing equations. In addition to the piecewise linear bottom, the model can also be run with full cells or partial cells (both full cell and partial cell options yield a staircase bottom that does not fit the real bottom topography). Two frictionless wave cases were chosen to evaluate the responses of the model to different treatments of the topography. One wave case is a boundary value problem, while the other is an initial value problem. To verify that the piecewise linear bottom does not cause increased diffusions for areas with steep bottom slopes, a barotropic case in a symmetric triangular basin was tested. The model was also applied to a real estuary using various topography treatments. The model results demonstrate that fitting the topography is important for the initial value problem. For the boundary value problem, topography-fitting may not be very critical if the vertical spacing is appropriate. Copyright © 2004 John Wiley & Sons, Ltd. [source] Study on Enzymatic Hydrolysis of Polylactic Acid by Endogenous Depolymerizaion ModelMACROMOLECULAR THEORY AND SIMULATIONS, Issue 6 2007Masaji Watanabe Abstract Enzymatic degradation of polylactic acid is studied experimentally and analytically. Gel permeation chromatography profiles obtained before and after the enzymatic degradation of polylactic acid (PLA) were introduced into the analysis based on a mathematical model. Previously developed techniques were successfully adapted to the analysis of an initial value problem consisting of an endogenous depolymerization model and an initial condition, and an inverse problem to determine the degradation rate for which the solution of the initial value problem also satisfies a final condition. Those problems were solved numerically and numerical results are introduced. Degradabilities of PLA and polyvinyl alcohol are compared. [source] Local C1 solutions to some non-linear PDE systemMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2006Emanuele Callegari Abstract We prove that the quasilinear initial value problem (1) has a unique, local in time, C1 solution, if the matrices Ai are diagonalizable and commute with each other. Copyright © 2006 John Wiley & Sons, Ltd. [source] Existence and non-existence of global solutions of a non-local wave equationMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2004Azmy S. Ackleh Abstract We study the initial value problem where with ,(x),0 and . We show that solutions exist globally for 0 A fractional splitting algorithm for nonoverlapping domain decomposition for parabolic problemNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2002Daoud S. Daoud Abstract In this article we study the convergence of the nonoverlapping domain decomposition for solving large linear system arising from semi-discretization of two-dimensional initial value problem with homogeneous boundary conditions and solved by implicit time stepping using first and two alternatives of second-order FS-methods. The interface values along the artificial boundary condition line are found using explicit forward Euler's method for the first-order FS-method, and for the second-order FS-method to use extrapolation procedure for each spatial variable individually. The solution by the nonoverlapping domain decomposition with FS-method is applicable to problems that requires the solution on nonuniform meshes for each spatial variable, which will enable us to use different time-stepping over different subdomains and with the possibility of extension to three-dimensional problem. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 609,624, 2002 [source] Aspects of vortex dynamics in Ginzburg-Landau modelsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007Fabrice Bethuel We survey some recent work concerning the asymptotic dynamics of vortices in the 2-dimensional parabolic Ginzburg-Landau equation, the interaction of vortices with the phase field and the limiting initial value problem for both vortices and phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A New Numerical Approach for a Detailed Multicomponent Gas Separation Membrane Model and AspenPlus SimulationCHEMICAL ENGINEERING & TECHNOLOGY (CET), Issue 7 2005M. H. Murad Chowdhury Abstract A new numerical solution approach for a widely accepted model developed earlier by Pan [1] for multicomponent gas separation by high-flux asymmetric membranes is presented. The advantage of the new technique is that it can easily be incorporated into commercial process simulators such as AspenPlusTM [2] as a user-model for an overall membrane process study and for the design and simulation of hybrid processes (i.e., membrane plus chemical absorption or membrane plus physical absorption). The proposed technique does not require initial estimates of the pressure, flow and concentration profiles inside the fiber as does in Pan's original approach, thus allowing faster execution of the model equations. The numerical solution was formulated as an initial value problem (IVP). Either Adams-Moulton's or Gear's backward differentiation formulas (BDF) method was used for solving the non-linear differential equations, and a modified Powell hybrid algorithm with a finite-difference approximation of the Jacobian was used to solve the non-linear algebraic equations. The model predictions were validated with experimental data reported in the literature for different types of membrane gas separation systems with or without purge streams. The robustness of the new numerical technique was also tested by simulating the stiff type of problems such as air dehydration. This demonstrates the potential of the new solution technique to handle different membrane systems conveniently. As an illustration, a multi-stage membrane plant with recycle and purge streams has been designed and simulated for CO2 capture from a 500,MW power plant flue gas as a first step to build hybrid processes and also to make an economic comparison among different existing separation technologies available for CO2 separation from flue gas. [source] Stress analyses of laminates under cylindrical bendingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2008Tarun Kant Abstract A semi-analytical approach for evaluation of stresses and displacements in composite and sandwich laminates under cylindrical bending subjected to transverse load has been developed in this paper. Two dimensional (2D) partial differential equations (PDEs) of such a laminate are obtained by imposing plane-strain conditions of elasticity. The fundamental dependent variables are so selected in this formulation that they satisfy the continuity of displacements and transverse interlaminar stresses at the laminate interface through the thickness. The set of governing PDEs are transformed into a set of coupled first-order ordinary differential equations (ODEs) in thickness direction by assuming suitable global orthogonal trigonometric functions for the fundamental variables satisfying the boundary conditions. These ODEs are numerically integrated by a specially formulated ODE integrator algorithm involving transformation of a two-point boundary value problem (BVP) into a set of initial value problems (IVPs). Numerical studies on both composite and sandwich laminates for various aspect ratios are performed and presented. Accuracy of the present approach is demonstrated by comparing the results with the available elasticity solution. It is seen that the present results are in excellent agreement with the elasticity solutions. Some new results for sandwich laminates and for uniform loading condition are presented for future reference. Copyright © 2006 John Wiley & Sons, Ltd. [source]
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