Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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duration of the simulated phenomenon, limit the range of problems that can be solved by MD methods. Although MD has advanced tremendously and gained in popularity on the wave of the extremely rapid development of computer technology, the current state the matter still makes unthinkable the macroscopic modeling of even the shortest physical phenomena. For example, MD modeling of a 1 [mm] 3 copper sample is not even closely achievable, because it consists of approximately 10 21 copper atoms, which exceeds by far the capabilities of even the most powerful computer systems at present. The size of the MD simulation cell is limited typically up to 10 8 atoms nowadays, which ordinarily corresponds to a few tens of nanometers. According to the available data, it appears that the current record of 4 . 1 · 10 12 atoms (equivalent to a 4 . 5 × 4 . 5 × 4 . 5 [ µ m ] cubic simulation cell) has been achieved on the computer platform “SuperMUC” (Leibniz Supercomputer Center of the Bavarian Academy of Sciences) using 131072 processors and over 500 TFLOPS. The simulation duration is defined by the size of the time step and the number of cycles. A typical time step in MD simulations is of the order of femtosecond. As an example, for 2D Leonard Jones systems, the initial time step estimate is based on the expression δ t = 1 60 ÷ 1 30 2 π ω E , ω E = √ 3 ω 0 (6.22) where ω E denotes the Einstein frequency associated with the fundamental harmonic fre quency ω 0 = C 0 / r 0 defined in terms of the speed of sound and the equilibrium interatomic distance [39]. The small time step required for MD simulations of atomic systems severely limits the total duration of the event simulated. By using modern computers and massively parallel processing it is possible to calculate approximately 10 8 time cycles (that is, to simulate physical phenomena that occur within 100 [ns]). This is a serious limitation for many problems involving thermally activated processes (Table 6.1, Table 6.2), which is why several methods have been developed for their acceleration for the purpose of studying surface diffusion, film deposition, and the evolution of point defects. A record MD simulation of the order of 10 [ µ s ] describes the rapid bending of protein chains [58]. Increasing the physical duration of simulated processes is a constant, active research area that includes the improvement of numerical algorithms, parallel processing, and the continuous development of hardware capabilities. Table 6.1: Examples of representative temporal and spatial scales necessary to observe some typical deformation mechanisms under step-pressure loading of single-phase metals (adopted from [59]).

Mechanism

Representative Length Scale Representative Time Scale

Phase transformation Dislocation nucleation

10 nm 50 nm

10 ps 50 ps

Twin formation

1 nm

1 ns

Interaction of dislocations Spallation; tensile damage

100 nm

100 ns 100 ns 10 µ s

1 µ m

100 µ m

Adiabatic shear