On efficient and reliable stochastic generation of RVEs for analysis of composites within the framework of homogenization

Abstract

In this paper we describe efficient methods of generation of representative volume elements (RVEs) suitable for producing the samples for analysis of effective properties of composite materials via and for stochastic homogenization. We are interested in composites reinforced by a mixture of spherical and cylindrical inclusions. For these geometries we give explicit conditions of intersection in a convenient form for verification. Based on those conditions we present two methods to generate RVEs: one is based on the random sequential adsorption scheme, the other one on the time driven molecular dynamics. We test the efficiency of these methods and show that the first one is extremely powerful for low volume fraction of inclusions, while the second one allows us to construct denser configurations. All the algorithms are given explicitly so they can be implemented directly.

Appendices

Appendix 1. Validation of dynamics

The Figs. 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 represent typical intersections of spheres and cylinders and the result of application of the relaxation procedure to them.

Fig. 15

Two spheres a intersecting b after relaxation

Fig. 16

A sphere and a cylinder, symmetric intersection, type $sc2$ a intersecting b after relaxation c intersecting, top view d after relaxation, top view

Fig. 17

A sphere and a cylinder, not symmetric intersection, type $sc2$—the cylinder is turning (compare with Fig. 16) a intersecting b after relaxation c intersecting, top view d after relaxation, top view

Fig. 18

A sphere and a cylinder, axially symmetric intersection with the base, type $sc2$ then $sc3$ a intersecting b after relaxation

Fig. 19

A sphere and a cylinder, intersection with the base, type $sc3$ a intersecting b after relaxation

Fig. 20

A sphere and a cylinder, intersection with the base boundary, type $sc4$ a intersecting b after relaxation

Fig. 21

Two cylinders, symmetric intersection, type $cc1$ a intersecting b after relaxation

Fig. 22

Two cylinders, not symmetric intersection, type $cc1$—one cylinder is turning a intersecting b after relaxation c intersecting, front view d after relaxation, front view

Fig. 23

Two cylinders, not symmetric intersection, type $cc1$—both cylinders are turning (compare with Fig. 22) a intersecting b after relaxation c intersecting, front view d after relaxation, front view

Fig. 24

Two cylinders, axes in the same plane, intersection with one base, type $cd1$ a intersecting b after relaxation

Fig. 25

Two cylinders, parallel axes, intersections of type $cc1$ (degenerate case) and $cd2$, then $cd1$ a Intersecting b After relaxation

Fig. 26

Two cylinders, axes are not coplanar, intersection with one base, type $cd1$ a intersecting b after relaxation

Fig. 27

Two cylinders, axes are not coplanar, intersection with one base, type $cd1$ a intersecting b after relaxation

Fig. 28

Two cylinders, intersection of bases, type $d1$ or $d2$ a intersecting b after relaxation

Fig. 29

Two cylinders, axes intersect inside the cylinders, type $cc1$ (degenerate case), then $cd2$, then $cd1$ a intersecting b after relaxation

Appendix 2. Time needed for generation using the RSA algorithm

The Tables (2, 3, 4, 5, 6) show the dependence of time of construction of the RVEs following the algorithm 5 on various parameters of them: volume fractions $f_s, f_c$, number of inclusions, their geometry. The couples of numbers in the cells of the tables correspond to two values of the aspect ratio $a$ of cylinders (ratio between its length and diameter). The time estimation (in seconds) is averaged over 20 runs.

Table 2 RSA: Average time of RVE generation (in seconds) for 10 spheres, 10 cylinders, values of aspect ratio $a = 3|5$

Table 3 RSA: Average time of RVE generation (in seconds) for 20 spheres, 20 cylinders, values of aspect ratio $a = 3|5$

Table 4 RSA: Aaverage time of RVE generation (in seconds) for 30 spheres, 30 cylinders, values of aspect ratio $a = 3|5$

Table 5 RSA: Average time of RVE generation (in seconds) for 40 spheres, 40 cylinders, values of aspect ratio $a = 3|5$

Table 6 RSA: Average time of RVE generation (in seconds) for 50 spheres, 50 cylinders, values of aspect ratio $a = 3|5$

Appendix 3. Time needed for generation using the MD method

The Tables (7, 8, 9, 10, 11) show the dependence of time of construction of the RVEs using the time-driven MD relaxation method on various parameters of them: volume fractions $f_s, f_c$, number of inclusions, their geometry. The couples of numbers in the cells of the tables correspond to two values of the aspect ratio $a$ of cylinders (ratio between its length and diameter). The time estimation (in seconds) is averaged over 20 runs.

Table 7 MD: Average time of RVE generation (in seconds) for 10 spheres, 10 cylinders, values of aspect ratio $a = 3|5$

Table 8 MD: Average time of RVE generation (in seconds) for 20 spheres, 20 cylinders, values of aspect ratio $a = 3|5$

Table 9 MD: Average time of RVE generation (in seconds) for 30 spheres, 30 cylinders, values of aspect ratio $a = 3|5$

Table 10 MD: Average time of RVE generation (in seconds) for 40 spheres, 40 cylinders, values of aspect ratio $a = 3|5$

Table 11 MD: Average time of RVE generation (in seconds) for 50 spheres, 50 cylinders, values of aspect ratio $a = 3|5$