Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Figure 6.4: Schematic representation of two different solid structures: (a) the body-centered cubic crystal lattice, and (b) a molecular chain. The latter indicates 2-atoms (chemical bond), 3-atoms (angle bending), and 4-atoms (torsion; dihedral) interactions typical of the intramolecular bonding interactions. The main feature of the MD method is the ability to analyze dynamics of (non)equilibrium processes with spatial resolution on the atomic scale. Thus, MD simulations play a role of computational microscope and have no computational alternative for many problems including atomic-scale phenomena that cannot be observed directly. MD can be considered a numerical simulation offshoot of statistical mechanics. It has found research application in a wide range of problems prevalent in various scientific fields; such as, for example: • Theoretical and statistical physics: fluid theory; properties of a statistical ensemble; structures and properties of small clusters; phase transitions,... • Materials science and mechanics of materials: point, linear and plane defects in crystals and their interactions; stable and metastable structures of complex alloys and related phase diagrams; amorphous materials; radiation damage to materials; microscopic damage and fracture mechanisms; surface reconstructions; melting; growth of thin films; friction,... • Biology, biochemistry and biophysics: molecular structure; chemical reactions; pro tein structure, functional mechanisms and folding process; drug design; vibrational relaxation and energy transfer; membrane structure; dynamics of biomolecules,... “Everything that living things do can be understood in terms of the jigglings and wigglings of atoms” [18]. The basic idea is simple. First, to setup the atomic system one must: i) define a set of initial conditions (initial positions r i and velocities v i of all atoms in the system), then ii) adopt the interatomic potential to define interatomic forces (internal forces), and finally iii) introduce (externally applied) load acting on the system. After that, the evolution of a system of atoms ( m i , r i , v i ) ( i = 1 , 2 , . . . , N ) is determined

6.2.1 Basic Idea of MD

by solving a system of equations of motion (6.1) 1 for each atom. The resulting force acting on each atom at a given moment f i j = | f i j | = − d Φ d r i j ; F i = − ∑ j f i j r i j r i j

(6.2)