This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, micro-fluidics, solids, interface problems, stochastic problems, and statistically self-similar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

We investigate the structural and mechanical properties of single-walled carbon nanotubes (SWNTs) under hydrostatic pressure, using constant-pressure molecular dynamics (MD) simulations. We observed that all the SWNTs, independent of their size and chirality, behave like a classical elastic ring exhibiting a buckling transition transforming their cross-sectional shape from a circle to an ellipse. The simulated critical transition pressure agrees well with the prediction from continuum mechanics theory, even for the smallest SWNT with a radius of 0.4nm. Accompanying the buckling shape transition, there is a mechanical hardness transition, upon which the radial moduli of the SWNTs decrease by two orders of magnitude. Further increase of pressure will eventually lead to a second transition from an elliptical to a peanut shape. The ratio of the second shape transition pressure over the first one is found to be very close to a constant of ∼1.2, independent of the tube size and chirality.

We propose a theoretical model of the double ionization of homonuclear diatomic targets by fast electron impact. By application of two-effective-center continuum waves to describe the ejected electrons in the exit channel and by use of Nordsieck-type integrals, an expression has been obtained by an analytical treatment for the fivefold differential cross section, the relevant quantity to describe the kinematically complete collisions in the coplanar geometry. The correlate motion of the ejected electrons is taken into account by means of adequate Sommerfeld parameters.

Outflow boundary conditions (OBCs) are investigated for calculation of incompressible flows by spectral element methods. Several OBCs, including essential-type, natural-type, periodic-type and advection-type, are compared by carrying out a series of numerical experiments. Especially, a simplified form of the so-called Orlanski's OBCs is proposed in the context of spectral element methods, for which a new treatment technique is used. The purpose of this paper is to find stable low-reflective OBCs, suitable and flexible for use of spectral element methods in simulation of incompressible flows in complex geometries. The computation is firstly carried out for a 2D simulation of Poiseuille-Bénard channel flow with Re=10, Ri=150 and Pr=2/3. This flow serves as a useful example to demonstrate the applicability of the proposed OBCs because it exhibits a feature of vortex shedding propagating through the outflow boundary. Then a 3D flow around an obstacle is computed to show the efficiency in the case of more general geometries. Among the tested OBCs, the advection-type OBCs are proven to have better behavior as compared with the others.

In this paper, we propose efficient algorithms for approximating particular solutions of second and fourth order elliptic equations. The approximation of the particular solution by a truncated series of Chebyshev polynomials and the satisfaction of the differential equation lead to upper triangular block systems, each block being an upper triangular system. These systems can be solved efficiently by standard techniques. Several numerical examples are presented for each case.

On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S = 1/2 was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. However, on these networks the Ising model spin S =1 was seen to show a spontaneous magnetisation. In this case, a first-order phase transition for values of connectivity z = 2 and z = 7 is well defined. On these same networks the Potts model with q=3 and 8 states is now studied through Monte Carlo simulations. We also obtained for q = 3 and 8 states a first-order phase transition for values of connectivity z = 2 and z = 7 for the directed Barabási-Albert network. Theses results are different from the results obtained for the same model on two-dimensional lattices, where for q=3 the phase transition is of second order, while for q=8 the phase transition is of first-order.

In this paper, we describe a modified RATTLE (M-RATTLE) method for rigid body dynamics directly in Cartesian coordinates. The M-RATTLE method introduces a new way of resetting the coordinates to satisfy the constraints at each step, which is designed for the rigid body dynamics calculations in the Cartesian coordinates. M-RATTLE is algebraically equivalent to the RATTLE method and the cost of performing rigid body dynamics by M-RATTLE is independent of the number of constraints. The interaction forces between atoms belonging to the same rigid molecule do not need to be computed and explicit expressions of the constraints of internal degrees of freedom are unnecessary. The performance and sampling results of the proposed method are compared with those of the symplectic splitting method for an isolated rigid benz molecule and for a cluster of twenty-seven benz molecules.

The need to perform spatial queries and searches is commonly encountered within the field of computational physics. The development of applications ranging from scientific visualization to finite element analysis requires efficient methods of locating domain objects relative to general locations in space. Much of the time, it is possible to form and maintain spatial relationships between objects either explicitly or by using relative motion constraints as the application evolves in time. Occasionally, either due to unpredictable relative motion or the lack of state information, an application must perform a general search (or ordering) of geometric objects without any explicit spatial relationship information as a basis. If previous state information involving domain geometric objects is not available, it is typically an involved and time consuming process to create object adjacency information or to order the objects in space. Further, as the number of objects and the spatial dimension of the problem domain is increased, the time required to search increases greatly. This paper proposes an implementation of a spatial k-d tree (skD-tree) for use by various applications when a general domain search is required. The skD-tree proposed in this paper is a spatial access method where successive tree levels are split along different dimensions. Objects are indexed by their centroid, and the minimum bounding box of objects in a node are stored in the tree node. The paper focuses on a discussion of efficient and practical algorithms for multidimensional spatial data structures for fast spatial query processing. These functions include the construction of a skD-tree of geometric objects, intersection query, containment query, and nearest neighbor query operations.