Next: Initialization Up: Basic DPD Equations Previous: Basic DPD Equations
The original DPD algorithm [Koelman and Hoogerbrugge (1993); Boek et al. (1997)] used a simple Euler algorithm for time integration. It has been noted, in Groot and Warren (1997), that use of a modified velocity-Verlet algorithm leads to improvements in numerical accuracy as well as a better characterization of thermal equilibrium properties for the DPD simulation [for a discussion of various integration schemes see Vattulainen et al. (2002)]. The original velocity Verlet algorithm [Verlet (1967)] is widely used in simulations and is an example of a second order symplectic integrator that has minimal computational memory requirements. It has the form
where a(0)
= F[x(0)]/m is the acceleration term evaluated using x(0) and
an intermediate velocity
(0). The
velocity Verlet algorithm does not provide a prescription for including
velocity dependent forces as found in DPD. To extend the velocity Verlet
algorithm to include velocity dependent forces we follow
Groot and Warren (1997), where a(0) =
F[x(0),(0)]/m and define
to be
To model rigid body motion in a fluid, a subset of the DPD particles are initially assigned a location in space such that they approximate the shape of the object. The motion of these particles is then constrained such that their relative positions never change. The total force and torque are determined from the DPD interparticle interactions and the rigid body moves according to the Euler equations for rigid bodies. The Euler equations were solved using a quaternion based computational approach proposed by Omelyan (1998). The details of this algorithm and its adoption for DPD are given in Martys and Mountain (1999).