Time-integration scheme Euler explicit

If a single-stage Euler explicit time-integration scheme is used, the updated moment set can be written as... 

Eq. (B.l). Thus, as a first step, we need to consider the volume-average form of Eq. (B.l) or, equivalently, the volume-average forms of the individual terms in Eqs. (B.2)-(B.5). Using a single-stage Euler explicit time-integration scheme (Leveque, 2002), the finite-volume expression for the updated NDF has the form ... 

The volume fraction variables were assumed to be time dependent in the semi-implicit discretization scheme used to solve the continuity equations. Considering the last two terms in the above relation, the transient term was then approximated by an explicit Euler time discretization scheme followed by volume integration in which the transient term was kept constant. If we thereafter multiply the resulting relation by Atjp, the relation can be rewritten as ... 

Let us now discuss in detail the question of moment conservation during time integration. Consistently with Chapter 8, the source terms due to phase-space processes are set to zero so that only transport terms in real space are considered in this discussion. When Eq. (D.23) is integrated using an explicit Euler scheme, the volume-average moment of order k in the cell centered at X at time (n + l)Af is directly calculated from the volume-average moment of order k at time n Af from the following equation ... 

Fig. 7.5. Rotation of a twisted spiral. Left Spatial distribution of Rcij. Right Position of the spiral at times t = 0 (solid), ( = 340 (dashed), t = 680 (dotted). Parameters are a = 4.19, / = 0.992, v = 3.9895, B = 0.045.5, the system siz.e is 500 x. 500. Numerical integration using the explicit Euler scheme with Ax = 0.2 and At = 0.0025.

Equations (1) and (2) are solved by Euler s explicit scheme using finite difference method. The acceleration of a particle can be obtained from the known contact forces, moments, mass and mass moment of inertia. The acceleration is integrated by time to deld the velocity increment and the velocity is integrated again by time to yield the displacement increment. By repeating these processes for all particles, the motion of all particles and the dynamical behavior of granular material can be obtained entirely. Consequently, the unbalanced forces and moments produce the linear and rotational accelerations of particle in the next calculating step successively. 

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