Method of Numerical Simulation

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

One can consider equations (3.37), (3.39) and (3.41) to be a basic system of equations for description of dynamics of entangled systems. The system can be investigated analytically in linear approximation as will be demonstrated in the ensuing chapters. However, to study these non-linear equations in complete form, one has to use numerical methods of simulation of the stochastic processes for the particle coordinates. 

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Tang, H.M., Yan, E.C., Hu, X.L., 2001. Theory and method of numerical simulation of engineering geology. Wuhan China University of Geosciences Press. 

In general, the research methods of debris flow pressure include theoretical analysis (Lin and Lin 1999 Yu and Tuan 2003), laboratory experimentation (Haukssona et al. 2007 Miyoshi and Suzuki 1990) and numerical simulation. In recent years, the methods of numerical simulation have been used to solve limit pressure against debris flow. For example, Teufelsbauer et al. (2011) presented a DEM model for simulating dry granular avalanche down an incline moreover, the flow pattern and pressure on rigid obstacles can be obtained. 

Finally, three appendices are provided, which recap a few notions about statistical methods of numerical simulation (Appendix 1), and offer some reminders about the properties of solutions (Appendix 2) and statistical thermodynamics (Appendix 3) - subjects which were discussed in detail in the first volume of this series. 

Among all the known methods of numerical simulation, we use the most versatile method of spectral collocation to analyze classical breathers in our system of ferroelectrics. This method is not only the latest numerical technique with ease of implementation, but also gives rise to a minimum of errors in the analysis. Spectral methods are a class of spatial discretizations for differential equations. In order to prepare the equation for numerical solution we introduce the auxiliary variable Q. = p, = — -. This reduces the second order Eq. (2) to the first order system ... 

In this paper we discussed the mixing ability of the normal screw element, kneading disc element and VCR element by the method of numerical simulation and experiment, the main results are ... 

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 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. 

The mental images, no matter how well grounded scientifically, are individually and collectively biased, as they have been developed after considerable filtering. The filters result from the scientific training of individuals, available supporting information from other processes, existing theoretical methods, limitations of numerical simulation, and characteristics of experimental methods. 

Baker et al. (1978a) developed a method which can predict blast pressures in the near field. This method is based on results of numerical simulations (see Section 6.3.1.1) and replaces Step 5 of the basic method (Figure 6.20). The refined method s procedure is shown in Figure 6.25. 

The most interesting theoretical problems in Earth system science cannot be solved by analytical methods their solutions cannot be expressed as algebraic expressions and so numerical solutions are needed. In this chapter I shall introduce a method of numerical solution that can be applied to a wide range of simulations and yet is easy to use. In later chapters I shall elaborate and apply this method to a variety of situations. 

The results of numerical simulation of bluff-body stabilized premixed flames by the PPDF method are presented in section 12.2. This method was developed to conduct parametric studies before applying a more sophisticated and CPU time consuming PC JVS PDF method. The adequate boundary conditions (ABC) at open boundaries derived in section 12.3 play an essential role in the analysis. Section 12.4 deals with testing and validating the computational method and discussing the mechanism of flame stabilization and blow-off. 

D. T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reac-... 

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