Simulation tree structure

In many gas networks, the net has a special tree structure, with one source (a supplier) and several sinks (consumers), as illustrated in figure 1. This simplifies the simulation of the gas net to a very large extent. A gas net of such a tree structure is common for the nitrogen purge systems in petrochemical plants, which is the practical example for this analysis. 

The simulation of a gas distribution system having a tree structure with one supplier may be reduced to a system of equations which has as many dependent variables as there are sinks or consumers. At each sink, the pressure (pressure control loop termination) or the mass flowrate (mass flowrate control loop termination) must be specified. The purpose of this short paper is to demonstrate some of the most commonly used mathematical models for the simulation of a gas net with a tree structure and point out some of the problons which arise if the net has an overload, a severe leakage or a bleeding at one point or another leading to choked flow. 

The computer time per reaction of this algorithms scales with system size as 0(log S) where S is the number of sites in the system. (Note that for all kMC algorithms the total number of reactions in a system is of the order 0(S). So for the first-reaction method the computer time for a whole simulation scales as 0(S log 5).) This logarithmic dependence originates from the data-structures, which are normally trees, that are used to store the reactions and their times. 

The constitutive equations use a thermodynamic framework, that in fact embodies not only purely mechanical aspects, but also transfers of masses between the phases and diffusion of matter through the extrafibrillar phase. Since focus is on the chemo-mechanical couplings, we use experimental data that display different salinities. The structure of the constitutive functions and the state variables on which they depend are briefly motivated. Calibration of material parameters is defined and simulations of confined compression tests and of tree swelling tests with a varying chemistry are described and compared with available data in [3], The evolution of internal entities entering the model, e.g. the masses and molar fractions of water and ions, during some of these tests is also documented to highlight the main microstructural features of the model. 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

One of the features of the forest fire problem making rigorous analysis difficult is the presence of individual trees, shrubs, bushes, trunks, branches, leaves and occasional human structure. To simulate these elements in detail with sufficient accuracy to replicate individual vortical motions would be intractable. Over a region of even a few acres there must be thousands of individual unequally sized and spaced objects. 

In each of the above analyses, the pores were considered as parallel sets of large and small pores without interconnection between the separate sets. However, most void structures comprise a network in interconnected void spaces and "network effects" will diaate the potential implications of changes in pore structure. The generic influence of pore networks were analyzed by Beeckman and Froment22 based on modified Bethe tree two-dimensional networks. Based on this simulated analyses, the authors concluded that the nature of the deactivation does depend on the nature of the network structure. Sahami and Tsotsis employed percolation theory to analyze a three-dimensional network of interconnected pores and concluded that the void interconnectivity is crucial in determining the influence of network structure on the deactivation phenomena. 

Equation (4.21) states that the dynamics of the forward-rate process, beginning with the initial rate/(0, J), are specified by the set of Brownian motion processes and the drift parameter. For practical applications, the evolution of the forward-rate term structure is usually derived in a binomial-type path-dependent process. Path-independent processes, however, have also been used, as has simulation modeling based on Monte Carlo techniques (see Jarrow (1996)). The HJM approach has become popular in the market, both for yield-curve modeling and for pricing derivative instruments, because it matches yield-curve maturities to different volatility levels realistically and is reasonably tractable when applied using the binomial-tree approach. 

The patterns produced by the diffusion-limited aggregation (DLA) processes are characterized by the open random and tree-type structures and can be well described as fractals. Computer simulations of fractal growth have been shown to produce structures... 

Mandelbrot [2, 3] systematized and organized mathematical ideas concerning complex structures such as trees, coastlines and non-equilibrium growth processes. He pointed out that such patterns share a central property and symmetry which may be called scale invariance. These objects are invariant under a transformation, which replaces a small part with bigger part that is under a change in a scale of the picture. Scale-invariant structures are called fractals [7]. More recently the relevance of natural and mathematical structure has become clearer with the help of computer simulation. Self-similarity turns out to be a general invariance principle of these structures. 

Construct the product alternating automaton = K x Ao,(f. This automaton simulates a run of Ao,(f on the tree induced by the Kripke structure K. 

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