Lineal-Path Function

The lineal-path function considers information on the connectedness of a phase and was first proposed as a statistical measure by Lu and Torquato [85]. This function captures the probability of a line segment −−→x1x2, with the start and end points given by the vectors x₁ and x2, being located entirely in phase P . Therefore, an indicator function is defined_{original rebinned}

original rebinned

(b) (a)

Figure 6.4: Schematic illustration of the rebinning procedure used on a 2D spectral density.

kx

ky 1 2

4 3

original

kx

ky

3 4

2 1

reordered

Figure 6.5: Schematic illustration of reordering scheme used on a 2D spectral density. The zero component is marked with a white square.

(a) periodic microstructure (b) unitcell

(c) rebinned and reordered spectral density of periodic microstructure

(d) reordered spectral density of unitcell

Figure 6.6: (a) Periodic microstructure and (c) its spectral density. (b) Periodic unitcell and (d) its spectral density.

by

χ^{(P )}_LP(−−−→x₁x₂) :=

(1 if −−→x₁x₂ ∈ D^{(P )},

0 otherwise. (6.21)

The lineal-path function is defined by the ensemble average over a series of samples α P_LP^{(P )}(−−−→x₁x₂) := χ^{(P )}_LP(−−−→x₁x₂, α), (6.22) whereas for ergodic microstructures the volume average over an infinite domain B can be considered instead of the ensemble average, hence

P_LP^{(P )}(−−−→x1x2) := lim

V(B)→∞

Z

B

χ^{(P )}_LP(y + −−−→x1x2) dy, (6.23) where y + −−−→x1x2 denotes a shift of the line −−−→x1x2 by the position vector y. Note again that the volume integral over an infinite volume is replaced by the integral over a sufficiently large volume for practical reasons. The lineal-path function has been used as a statistical descriptor in, e.g., Yeong and Torquato [176], where three-dimensional microstructures are reconstructed based on two-dimensional cuts using different combi-nations of statistical measures. Havelka et al. [53] propose an accelerated method for the computation of lineal path functions in the reconstruction of random microstructures using a parallelization of its computation.

The computation of a complete lineal-path function would require an analysis of all pos-sible line segments with varying length and orientation which is a tremendous computa-tional effort even for reasonably sized microstructures and increases with higher dimen-sions. Therefore, sampling techniques have been developed which lower the computational expenses. In Seaton and Glandt [136], a sampling technique based on Monte-Carlo method was proposed for the computation of spatial correlation functions. In application to the lineal-path function, this can be understood as an evaluation of a large number of line segments which are randomly thrown into a medium. Another approach presented in Smith and Torquato [144] uses a sampling template for the evaluation of two-point correlation functions. In Zeman [179], the sampling template is constructed for lineal-path functions, comprising a fixed number of line segments. The template is a discrete field where entries marked with the value 1 represent the end point of a line starting from the center. All other entries are zero. In order to construct such templates, the digitalization of line elements can be carried out using suitable algorithms, as given in Bresenham [24]. The size of the template is defined by Tx = 2 Nx − 1, Ty = 2 Ny − 1 and Tz = 2 Nz − 1 for a three-dimensional discrete medium of the size Nx, Ny and Nz. Thereby, it comprises all possible line length which can be placed in the medium. Since the lineal-path function is invariant with respect to reflections of line segments, it is ad-missible to neglect the reflection of line segments in one direction, such that the template size reduces to Tx = 2 Nx− 1, Ty = 2 Ny− 1 and Tz = Nz. An example of such a template for the three dimensional case is shown in Fig. 6.7. All line segments start start from the center point. For the discrete case, the computation of the lineal-path function can then be carried out evaluating

for the two-dimensional case and

P_LP^{(P )}(m, k, l) := 1 for the three-dimensional case. The line segment is defined by the start and end points x1 = [p, q]^T and x2 = [m, k]^T in 2D and x1 = [p, q, r]^T and x2 = [m, k, l]^T in 3D, respectively. ˜N is the number of admissible discrete points in the range of the summation limits. The summation limits are given by

pm = max[0, −m], pM = min[Nx, (Nx− m)] , qk = max[0, −k], qK = min[Ny, (Ny − k)] , rl = max[0, −l], rL = min[Nz, (Nz− l)] ,

(6.26)

thus ˜N = (pM − 1 − pm)(qK − 1 − qk)(rL− 1 − rl) in the three-dimensional case. The computation of the lineal-path function is carried out by placing the template with the center point in a selected point in the admissible range p ∈ [pm, pM] and q ∈ [qk, qK], which has to belong to the considered phase P . All points of the line segment are then checked for their phase affiliation. By summing over all admissible points, i.e. points inside the limit given in Eq. (6.26), and normalizing by ˜N , the lineal-path function can be computed.

Alternatively, the template can be placed at random points using Monte-Carlo method, where normalization is performed over the number of random placements.

m Nz

k

2Ny− 1 l

2Nx− 1

Figure 6.7: Reduced template for the computation of the 3D lineal-path function.

For periodic media, where only a periodic unitcell has to be analyzed, Eq. (6.24) and Eq. (6.25) can be reformulated to

P_LP^{(P )}(m, k) := 1 N˜

Nx

X

0 Ny

X

0

χ^{(P )}(−−−→x1x2) (6.27)

and

P_LP^{(P )}(m, k, l) := 1 N˜

Nx

X

0 Ny

X

0 Nz

X

0

χ^{(P )}(−−−→x₁x₂), (6.28) respectively, where lines continuing outside of the limits of the data set “reenter” the set of the respective other side due to periodicity. For a reduction of computational effort, the number of analyzed lines in the template can be altered as well. Therein, this reduction implies a reduction of the amount of information which is captured, which should to be done in a reasonable sense of the application.

Fig. 6.8 shows the lineal-path function for the DP steel microstructure analyzed in this work, where only a relevant section with a distinct probability threshold is shown. The lineal-path function enables the identification of an average inclusion size with a definition of relevant threshold p^thresof the probability of a line segment being located in the analyzed phase, this will be further discussed in Section 6.2.7.