Simulation numerical

A tsunami starts with a complicated initial profile, is transformed by the effects of complicated bottom topography, and runs up and down on land of complicated topography. In estimating the tsunami hazard, it is the usual practice to carry out a numerical simulation. In order to ensure stable computation and accurate results, dedicated validation programmes should be used. 

The examples we have studied thus far have had rather simple kinematics flow parallel or nearly parallel to a wall and ideal or nearly ideal extension. Thus, we have been able to obtain exact solutions for the flow or to obtain approximate solutions based on the small difference between the actual flow and an ideal case for which an exact solution is available. Even for the case of fiber spinning, where an analytical solution to the thin filament equations cannot be obtained under conditions relevant to industrial practice, we simply need to obtain a numerical solution to a pair of ordinary differential equations, which is a task that can be accomplished using elementary and readily available commercial software. 

The flow in many real processing geometries is too complex for us to apply the analytical methods utilized in the preceding chapters. Indeed, even when the flow field is a simple one, the coupled heat transfer problem may not be amenable to a simple treatment the elementary extruder in Chapter 3 is an example of a case in which we are unable to obtain an exact or even approximate solution for the spatial development of the two-dimensional temperature held. 

Complex coupled flow and heat transfer problems can be solved using numerical techniques in which the partial differential equations are converted to a large set of coupled algebraic equations, and the algebraic equations are then solved using conventional methods developed specificaUy to be efficient on digital computers. The concept by which the numerical solution of the partial differential equations is obtained is rather straightforward, and we wiU describe it here. Actual implementation into an efficient, user-friendly computer code is difficult and tedious, however, and most users employ commercial software. 

Most modern computer codes for low Reynolds number flow, the case in which we are generally interested in polymer processing applications, use an approximation 

As an example of the application of Galerkin s method, consider the following linear boundary-value problem  

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The development of Remote Field Eddy Current probes requires experience and expensive experiments. The numerical simulation of electromagnetic fields can be used not only for a better understanding of the Remote Field effect but also for the probe lay out. Geometrical parameters of the prohe can be derived from calculation results as well as inspection parameters. An important requirement for a realistic prediction of the probe performance is the consideration of material properties of the tube for which the probe is designed. The experimental determination of magnetization curves is necessary and can be satisfactory done with a simple experimental setup. 

The flow processes are described as Marangoni convections and up to now they were determined by several research centers through numeric simulation works [9]. Due to the... 

As these examples have demonstrated, in particular for fast reactions, chemical kinetics can only be appropriately described if one takes into account dynamic effects, though in practice it may prove extremely difficult to separate and identify different phenomena. It seems that more experiments under systematically controlled variation of solvent enviromnent parameters are needed, in conjunction with numerical simulations that as closely as possible mimic the experimental conditions to improve our understanding of condensed-phase reaction kmetics. The theoretical tools that are available to do so are covered in more depth in other chapters of this encyclopedia and also in comprehensive reviews [6, 118. 119],... 

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

Figure A3.14.17. Self-replicating spots in the FIS reaction in a CFUR, comparing an experimental time sequence with numerical simulation based on a simple autocatalytic scheme. (Reprmted with pennission from Lee etal [M], Macmillan Magazines Ltd. 1994.)...

In the context of molecular simulation, particularly biomolecular modelling, a critical aspect for numerical simulation is the presence of long-range Coulombic forces which render the force computations much more costly... 

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. 

Mitsoulis, E., 1986. The numerical simulation of Boger fluids a viscometric approximation approach. Polym. Eng. Sci. 26, 1552-1562. 

Mitsoulis, E., 1990. Numerical Simulation of Viscoelastic Fluids. In Encyclopaedia of Fluid Mechanics, Vol. 9, Chapter 21, Gulf Publishers, Houston. 

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

Crochet, M. J., 1982, Numerical simulation of die-entry and die-exit flow of a viscoelastic fluid. In Numerical Methods in Forming Processes, Pineridge Press, Swansea. 

Beaulne, M. and Milsoulis, E. 1999. Numerical simulation of the fihn casting process. Int. Poly. Proce.ss. XfV, 261-275. 

Hannart, B. and Hoplinger, E.J., 1998. Laminar flow in a rectangular diffuser near Hele-Sliaw conditions - a two dinien.sioiial numerical simulation. In Bush, A. W., Lewis, B. A. and Warren, M.D. (eds), Flow Modelling in Industrial Processes, cli. 9, Ellis Horwood, Chichester, pp. 110-118. 

P. S. Gough, Numerical Simulation of Current Artilley Charges Using the TDNOVA Code, BRE-TR-555, BRL, Aberdeen, Md., June 1986. 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

Zannetti, Paolo, Numerical Simulation Modeling of Air Pollution An Oveiview, Ecological Physical Chemistiy, 2d International Workshop, May 1992. 

The methods discussed in the technical hterature are not exact. Numerical simulations of plant performance show that gross errors frequently remain undetected when they are present, or measurements are isolated as containing gross errors when they do not contain any. 

The present book, with contributions from a group of very knowledgable scientists in the field, is an attempt to provide a basis for addressing Bridgman s concerns. The response requires multidisciplinary contributions from solid mechanics, solid-state physics, materials science, and solid-state chemistry. Certainly, advances in theory, experimentation, and numerical simulation are impressive, and many aspects of shock-compressed solids have been studied in detail. At the fundamental level, however, it is certainly appropriate to question how well shock-compression processes are understood. 

Some wave phenomena, familiar to many people from the human senses, include the easy undulation of water waves from a dropped stone or the sharp shock of the sonic boom from high-speed aircraft. The great power and energy of shock events is apparent to the human observer as he stands on the rim of the Meteor Crater of Arizona. Human senses provide little insight into the transition from these directly sensed phenomena to the high-pressure, shock-compression effects in solids. This transition must come from development of the science of shock compression, based on the usual methods of scientific experimentation, theoretical modeling, and numerical simulation. 

Numerical simulation of a complex dynamic fracture application can be illustrated by calculations of impact induced damage in a ceramic cylinder. The computer model used was originally developed for oil shale explosive fragmentation (Grady and Kipp, 1980), with various extended applications considered by Boade et al. (1981) and Chen et al. (1983). In this model, stress and strain are related through... 

In this section, we discuss the role of numerical simulations in studying the response of materials and structures to large deformation or shock loading. The methods we consider here are based on solving discrete approximations to the continuum equations of mass, momentum, and energy balance. Such computational techniques have found widespread use for research and engineering applications in government, industry, and academia. 

Numerical simulations offer several potential advantages over experimental methods for studying dynamic material behavior. For example, simulations allow nonintrusive investigation of material response at interior points of the sample. No gauges, wires, or other instrumentation are required to extract the information on the state of the material. The response at any of the discrete points in a numerical simulation can be monitored throughout the calculation simply by recording the material state at each time step of the calculation. Arbitrarily fine resolution in space and time is possible, limited only by the availability of computer memory and time. 

Numerical simulations are designed to solve, for the material body in question, the system of equations expressing the fundamental laws of physics to which the dynamic response of the body must conform. The detail provided by such first-principles solutions can often be used to develop simplified methods for predicting the outcome of physical processes. These simplified analytic techniques have the virtue of calculational efficiency and are, therefore, preferable to numerical simulations for parameter sensitivity studies. Typically, rather restrictive assumptions are made on the bounds of material response in order to simplify the problem and make it tractable to analytic methods of solution. Thus, analytic methods lack the generality of numerical simulations and care must be taken to apply them only to problems where the assumptions on which they are based will be valid. 

B. Engquist and A. Majda, Absorbing Boundary Conditions for the Numerical Simulation of Waves, Math. Comput. 31, No. 139 (1977). 

The algorithm that is usually employed to account for the hydrogen positions is SHAKE [15,16] (and its variant RATTLE [17]). Stated in a simplistic way, the SHAKE algorithm assumes that the length of the X — H bond can be considered constant. Because in a numerical simulation there are always flucmations, this means that the deviation of the current length dt(t) of the Mi bond from its ideal (constant) bond length d° must be smaller than some tolerance value e. 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

The MathWorks Inc. (1993) SIMULINK Numerical Simulation Software - Reference Guide, The MathWorks Inc., Natick, Mass. 

Recently, Langer (1999) has joined the debate. He at first sounds a distinct note of scepticism ... the term numerical simulation makes many of us uncomfortable. It is easy to build models on computers and watch what they do, but it is often unjustified to claim that we learn anything from such exercises. He continues by examining a number of actual simulations and points out, first, the value of... 

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