To gain a fuller understanding of material properties, modelling at several length scales should be combined (often referred to as ``multiscale modelling''). Typically, experiments or simulations at one length scale suggests interesting phenomena to study at a shorter length scale, the result of which is used as parameters in models at a longer length scale. In the following we will give some examples, followed by some examples of ``hybrid'' simulation techniques, where several simulation paradigms are combined in a single simulation.
Dislocation level simulations require the knowledge of details of the short-range interactions between dislocations. Many of these can come from atomic-scale simulations. This can be done by setting up a very large system, and observe the dislocation processes in this system. Bulatov et al. (1998) have studied the behaviour of the cloud of dislocations emitted from a crack in an f.c.c. metal, and observed the creation and subsequent destruction of a Lomer-Cotrell lock. They propose to extract parameters for dislocation level simulations from such atomistic simulations. Schiøtz, Jacobsen, and Nielsen (1995) have observed a new dislocation multiplication mechanism that may be active under extremely large strain rates.
If the main goal of the simulation is to provide knowledge of a specific dislocation process, a more specialised simulation setup may be advantageous. Zhou, Preston, Lomdahl and Beazley (1998) have simulated the formation of a jog through the intersection of two extended dislocations. The simulations show the details of the interaction and provides an estimate of the involved critical stresses.
Recently an advanced simulation technique, the ``Nudged Elastic Band (NEB) Method'' (Mills, Jónsson and Schenter 1995) was used to study cross slip of screw dislocations at the atomic scale (Rasmussen, Jacobsen, Leffers, Pedersen, Srinivasan and Jónsson 1997; Pedersen, Carstensen and Rasmussen 1998). The NEB method identifies the entire transition path and can thus be used to find the energy of the transition state. This makes the method very suitable for studying elementary dislocation reactions, and can provide good input for dislocation level simulations. The group is presently using the technique to study dislocation annihilation (Pedersen et al. 1998).
A number of simulation techniques have been developed, where several simulation techniques are combined in a ``hybrid'' simulation. Typically, the idea is to describe ``interesting'' parts of the simulation atomistically, while other parts of the system are described by a model based on linear elasticity theory. The rationale behind this is that in most large-scale simulations, the majority of the atoms do not participate actively in the processes being studied, their main role is to propagate the elastic fields. In these regions of the simulation the strains are typically small, and linear elasticity is a good description. At the same time an atomistic description is required in dislocation core and other parts of the system, where non-linearities play a role.
The simplest way of doing this is by dividing the simulations into two zones. In the inner zone an atomistic description is used. The inner zone is surrounded by a much larger outer zone, described by elasticity theory typically using a continuous finite element method (Kohlhoff, Gumbsch and Fishmeister 1991; Gumbsch 1995). In this way the long-range elastic field is described in a computationally inexpensive way, and the atoms in the atomistic region see the correct response from the surrounding material. Great care must be taken in matching the two regions.
A related method consists of using a lattice Green's function to describe the linear region (Thomson, Zhou, Carlsson, and Tewary 1992; Canel, Carlsson, and Thomson 1995; Schiøtz and Carlsson 1997). In this method the atoms in the outer zone are considered to interact through linear spring forces, and the force matrix is inverted giving a lattice Green's function. The Green's function thus describes the displacement of the lattice at one point resulting from a force acting on the lattice at another point. The degrees of freedom associated with the atoms in the outer zone (typically a few million atoms) can then be eliminated, and only the degrees of freedom associated with the atoms in the inner zone need to be retained (typically less than 1000), together with the elements of the Green's function matrix connecting atoms in the inner zone. The method has been used to study dislocation emission from sharp and blunt cracks (Zhou, Carlsson, and Thomson 1993,1994; Schiøtz, Canel, and Carlsson 1997). A similar technique was used by Rao, Hernandez, Simmons, Parthasarathy and Woodward (1998) to study dislocation core structures.
The above-mentioned techniques have the disadvantage that the system must a priori be divided into two ``zones'', and dislocations and cracks are confined to the inner zone. An adaptive technique has been developed (Tadmor, Ortiz and Phillips 1996; Shenoy, Miller, Tadmor, Rodney, Phillips and Ortiz, to be published). In this method the system is described by a finite element method, but instead of an ordinary constitutive relation, the energy is calculated from a tiny atomistic simulation. Where necessary, the finite element mesh is automatically refined, eventually continuing until it is so fine that each element only contains one atom. In this limit the simulation converges to an ordinary atomistic simulation. The method is adaptive, so the mesh is automatically refined in areas where a detailed description is required, and coarsened where less detail is required. The method has been used to study the interactions of dislocations and cracks with grain boundaries (Shenoy, Miller, Tadmor, Phillips and Ortiz 1998; Shenoy et al., to be published).