Simulation 4 Numerical Methods

Okunbor, D.I., Skeel, R.D. Canonical numerical methods for molecular dynamics simulations. J. Comput. Chem. 15 (1994) 72-79. 

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. 

In general, discontinuities constitute a problem for numerical methods. Numerical simulation of a blast flow field by conventional, finite-difference schemes results in a solution that becomes increasingly inaccurate. To overcome such problems and to achieve a proper description of gas dynamic discontinuities, extra computational effort is required. Two approaches to this problem are found in the literature on vapor cloud explosions. These approaches differ mainly in the way in which the extra computational effort is spent. 

In the earliest applications of numerical methods for the computation of blast waves, the burst of a pressurized sphere was computed. As the sphere s diameter is reduced and its initial pressure increased, the problem more closely approaches a point-source explosion problem. Brode (1955,1959) used the Lagrangean artificial-viscosity approach, which was the state of the art of that time. He analyzed blasts produced by both aforementioned sources. The decaying blast wave was simulated, and blast wave properties were registered as a function of distance. The code reproduced experimentally observed phenomena, such as overexpansion, subsequent recompression, and the formation of a secondary wave. It was found that the shape of the blast wave at some distance was independent of source properties. 

A new chapter (5) on reaction intermediates develops a number of methods for trapping them and characterizing their reactivity. The use of kinetic probes is also presented. The same chapter presents the Runge-Kutta and Gear methods for simulating concentration-time profiles for complex reaction schemes. Numerical methods now assume greater importance, since useful computer programs are available. The treatment of pH profiles in Chapter 6 is much more detailed. 

To examine the details of the structure of flames in channels under quenching conditions, numerical methods were used. Two-dimensional CFD simulation of a propane flame approaching a channel between parallel plates was carried out using the FLUENT code [25]. The model reproduced the geometry of the real channels investigated experimentally. Close to the quenching limit, the burning velocity, dead space, and radius of curvature of the flames were all close to the experimental values. 

Again, these functional relationships should ideally be available in an explicit form in order to ease the numerical method of solution. Two-solute batch extraction is covered in the simulation example TWOEX. 

A summary of ISIM commands is found below. The ISIM manual contains more details on writing models in ISIM and on the numerical methods that are used in the simulations. It can be obtained from Prof. John L. Hay, ISIM International Simulation Limited, Technology House, Salford University Business Park, Lissadel Street, Salford M6 6AP, England, (Tel +44-(0)61-745 7444 Fax +44-(0)61-737 7700). 

Chapter 2 is employed to provide a general introduction to signal and process dynamics, including the concept of process time constants, process control, process optimisation and parameter identification. Other important aspects of dynamic simulation involve the numerical methods of solution and the resulting stability of solution both of which are dealt with from the viewpoint of the simulator, as compared to that of the mathematician. 

The simulation of heat transfer in a PCM in simple geometry as well as in a whole storage as described above can be done with many mathematical and engineering software tools like MathCad and EES. Better results can be achieved with commercial CFC solutions like FLUENT that only require very basic knowledge on numerical methods for heat transfer. 

A numerical method to simulate the performance of the storage with the PCM module was implemented using an explicit finite-difference method. The discretization of the model can be seen in Figure 145. 

More complicated numerical methods, such as the Runge-Kutta method, yield more accurate solutions, and for precisely formulated problems requiring accurate solutions these methods are helpful. Examples of such problems are the evolution of planetary orbits or the propagation of seismic waves. But the more accurate numerical methods are much harder to understand and to implement than is the reverse Euler method. In the following chapters, therefore, I shall show the wide range of interesting environmental simulations that are possible with simple numerical methods. 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

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