The adaptive time integration scheme

Finally, in order to enforce the microstructure periodicity (see Figure 3.14), the updated position of the particle is determined as

x_i(t+∆t)=x⁰_i(t+∆t)−¡ lRVE¢

inlint

Ãx⁰_i(t+∆t)

¡l_RVE¢

i

!

, i=1,...,n_dim, (3.12)

wherex⁰is the position obtained from Equation (3.8), (lRVE)i is the length of the RVE ini-th dimension (see Figure 3.8) and nlint(x) denotes the nearest lower integer rounding function.

Critical time step

It is well-known that explicit algorithms are, in general, conditionally stable. Therefore, having adopted the explicit Verlet integration scheme, it must be ensured that the time integration step does not surpass the critical stability value. If this happens, the solution becomes unstable and the average overlap residue does not decay, preventing the legal configuration from being achieved. Although the time is only interpreted as a pseudo-time in the current framework (i.e., its absolute value does not yield any physical significance), it is, of course, desirable to increase the integration time step to reach the legal configuration as fast as possible.

It can be argued that an implicit time integration algorithm could be alternatively adopted, taking advantage of the associated unconditional stability. However, an implicit algorithm would require the computation of the interaction forces multiple times in each iteration of the numerical procedure employed to solve the system of dynamic equilibrium equations. On the other hand, the adoption of more accurate explicit methods, such as the fourth-order Runge-Kutta, seems unnecessary as the increase in accuracy would not lead to any tangible benefit in the proposed scheme.

In this scenario, a robust way of estimating the critical time integration step must be devised.

To this end, it is convenient to make use of results in the field of discrete element methods (DEM), where the Verlet integration scheme goes by the name of explicit finite difference method (O’Sullivan and Bray (2004), Burns and Hanley (2017)). This is done with the understanding that the use of an isokinetic thermostat leads to an approach that is, in practice, slightly different from the standard Verlet scheme, as the velocity of the particles is rescaled between iterations to enforce the prescribed temperature. It also bears saying that DEM and molecular dynamics are very similar schemes, with the main difference being the more diverse shapes of particles and often the consideration of the rotational degrees of freedom in DEM.

The critical time step using the central finite differences is well-known and given by (Belytschko et al., 1985)

∆tcrit= 2

ω_max, (3.13)

where the maximum angular frequency,ωmax, can be computed as ωmax=p

λmax. (3.14)

Given the equations of dynamic equilibrium of an undamped system,

M¨x+K x=F, (3.15)

wherexand ¨xare the position and acceleration of the particles, F is the vector of external forces,Mis the mass matrix andKis the stiffness matrix,λmaxis the maximum eigenvalue of the amplification matrix,T, defined as

T=M⁻¹K. (3.16)

Consider a system of particles with similar massmand where the springs characterizing the interaction forces have the same stiffnessk. It can be shown that the critical time step in these conditions is always proportional top

m/k (Burns and Hanley, 2017). In O’Sullivan and Bray (2004), an approach inspired in FEM is used to determine the critical time step for a diverse number of particle arrangements. Each particle contact is considered as a strut finite element, with a tangential and a normal stiffness. The mass matrix of each strut element is defined as diagonal, similar to the ‘lumped’ mass matrices used in FEM. Its non-zero elements are determined by dividing the mass of the particle equally by its different contacts, i.e., the different finite elements connecting to the particle. Considering similar normal and shear stiffness values, O’Sullivan and Bray (2004) derive the critical time steps for several particle arrangements (e.g., two disks, the maximum packing of disks in R², cubic arrangement of spheres in R³) from the eigenvalues of the amplification matrix of the contact elements. These yield the following general result

∆tcrit= s 2

nc

sm

k , (3.17)

wherenc is the number of contacts or coordination number of the particles. Given that all spatial arrangements are considered symmetric and periodic, the coordination number is equal for all particles. Note that this result follows from the definition of the mass matrix of the strut element, where the mass of the particle is equally divided by all the contacts.

In the most general case of a system comprising different particle types and sizes, the derivation of such relations is infeasible. Therefore, in an attempt to take advantage of the valuable results in O’Sullivan and Bray (2004), it is here proposed that the critical time step is approximated by

∆tcrit≈α

s 2

max( ¯n_c,1) sm¯

k, (3.18)

where ¯n_cis the mean coordination number of the particles in the system, and ¯mis the mean of the particles’ mass. These averages are employed to account for the diversity in the number of neighbors and sizes of the different particles in the RVE. In turn,α≤1 can be interpreted as a safety factor multiplier that allows for a more conservative estimate of the critical time step.

Nonetheless, it is emphasized that α=1 has been successfuly adopted in all the performed simulations throughout this thesis and no signs of instability arised.

To compute the mean of the particles’ mass, ¯m, the mass of each particle is, without any loss of generality, set equal to the radius of the corresponding circumscribed circumpherence or sphere. Given that it is not sought to simulate a real physical system, such choice is not unique, being this approach adopted due to its computational simplicity and its ability to capture the difference in size between interacting particles.

In the present contribution, where the magnitude of the interaction forces is set equal to the intersection length,k=1 for every pair of particles. Concerning the nonlinear spring for disks/spheres of the same size², it follows from the differentiation of Equation (3.3) that

2It is remarked that the nonlinear interaction force law (see Equation (3.3)) is solely employed to perform a comparison between linear and nonlinear springs in the case of disks/spheres of the same size.

k(dm)=p Ãdm

2r

!p−1

. (3.19)

If general nonlinear springs (p>1) are considered, the stiffness k can be approximated by replacingr with ¯r, where ¯r is the average circumscribed disks/spheres radius, anddm with 2 ¯r−∆tv¯, where ¯vis the average velocity of all particles.

The proposed estimate for the critical time step deserves some comments. In the first place, from a physical point of view this approximation agrees with the fact that the coordination number dictates the number of non-zero elements in the global stiffness matrix and thus influences its overall stiffness. In the second place, when the system is near a legal configuration, most particles are in contact with one or no adjacent particles, being the critical time step derived for two disks/spheres in O’Sullivan and Bray (2004) recovered. Finally, the mean coordination number, ¯nc, can be efficiently determined in the proposed approach through the Verlet lists of all particles. The numerical results presented in Section 3.5 demonstrate the robustness of the proposed approach for several cases, being shown that Equation (3.18) is indeed a reasonable estimate of the critical time step even without considering the safety factor multiplier (α=1).