Algorithm, modification

New method of fracture propagation simulation is proposed. The method unites three sub-models that describe three (except the fluid filtration) processes that affect the fracture propagation. Important advance of the methodic is its ability to replace any sub-model without nmnerical algorithm modification. So the appropriate sub-model can be chosen for each process depending on the problem features. 

One of the genetic algorithm modifications is implemented in Mathcad as a user function MGA (Sabanin V. R. et al.). Its description can be found in literature. A real-life demonstration of the functimi is available online on Mathcad CalculatiOTi Server (http //twt.mpei.ac.ru/MASAV orksheets/Minimum.mcd). 

In practice modifications are made to incorporate thermostats or barostats that may destroy the time-reversible and symplectic properties. While extended-system algorithms such as Nose dynamics [41] can be designed on the principles of the reversible operators, methods that use proportional velocity or coordinate scaling [42] cannot. Such methods arc very... 

The equations of motion are integrated using a modified velocity Verlet algorithm. The modification is required because the force depends upon the velocity the extra step involves... 

Modifications to the basic Verlet scheme have been proposed to tackle the above deficiencies, particularly to improve the velocity evaluation. One of these modifications is the leap-frog algorithm, so called for its half-step scheme Velocities are evaluated at the midpoint of the position evaluation and vice versa [12,13]. The algorithm can be written as... 

The algorithm outlined above is a level 1 cell multipole or Cartesian multipole algorithm [28]. A number of modifications are possible. Accuracy can be raised by using higher order expansions, which unfortunately are more expensive. The cost can be alleviated by... 

The learning element consists of a Performance Index (PI) table combined with a rule generation and modification algorithm, which creates new rules, or modifies existing ones. The structure of a SOFLC is shown in Figure 10.17. With SOFLC it is usual to express the PI table and rulebase in numerical, rather than linguistic format. So, for... 

The MC method can be implemented by a modification of the classic Metropolis scheme [25,67]. The Markov chain is generated by a three-step sequence. The first step is identical to the classic Metropolis algorithm a randomly selected molecule i is displaced within a small cube of side length 26r centered on its original position... 

The other global method in common use is generally called the method of Qraejfe in western Europe and in the United States, and the method of Lobachevskii in the USSR. The first to state the principle was Dandelin, but Graeffe devised the algorithm, simple enough in itself, that is normally used. The method is widely known, and is described in many places. Hence, it will not be described here. But it is not widely known that the method is applicable, with rather trivial modifications, to solving transcendental equations. 

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