Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Figure 6.8 illustrates the Maxwell-Boltzmann velocity intensity distribution for tung sten (W) at temperatures of 300 [K] and 1000 [K]. As can be seen, the distribution is not symmetric: the average value of the velocity intensity (6.18) 1 is always greater than the most probable value (6.18) 2 (corresponding to the maximum of the distribution curve) ⟨ v ⟩ = r 8 k B T 0 π m , v m = r 2 k B T 0 m = √ π 2 ⟨ v ⟩ . (6.18)

Figure 6.8: The Maxwell-Boltzmann speed distributions for tungsten for two different temperatures.

6.2.5 MD Simulation Cell and List of Neighbors The response of small systems can be dominated by surface effects that may obscure the physical response of the bulk material. A simulation cell with periodic boundary conditions (Figure 6.9) is introduced to eliminate these contaminating surface effects whenever it is of interest to study behavior of the bulk material. The volume (surface area in 2D) of a periodic cell is representative of bulk material in the sense that it is considered to be composed of periodic cells surrounded on all sides by their exact replicas [5] as illustrated in Figure 6.9. This periodicity implies that the atom, in the bottom-right cell corner of Figure 6.9, leaving the MD simulation cell instantaneously reappears at the bottom-left cell corner. This elimination of unwanted surface effects using these periodic cells is achieved at the cost of introducing non-physical periodicity into the atomic system. Adverse consequences include: (i) unrealistically rigid response, (ii) unnatural wavelengths in the solution fields, (iii) suppression of localization that might otherwise occur [47], and (iv) violation of the conservation of angular momentum [15]. In order to simulate the most general loading conditions, it is necessary to be able to change the shape and size of the periodic cell and maintain constant values of certain state parameters (typically, temperature or pressure; the NTP ensemble). The restriction that the shape of the periodic cell must remain unchanged has a negative impact on the applicability