Computational optimization strategies for the simulation of random media and components

Abstract

In this paper efficient computational strategies are presented to speed-up the analysis of random media and components. In particular, a Hybrid Stochastic Optimization (HSO) tool, based on the synergy between various algorithms, i.e. Genetic Algorithms, Simulated Annealing as well as Tabu-list is suggested to reconstruct a set of microstructures starting from probabilistic descriptors. The subsequent analysis (e.g. Finite Element analysis) can be performed to obtain the desired macroscopic quantity of interest and, providing a link between the micro- and the macro-scale. Different computational speed-up strategies are also presented.

The proposed simulation approach is highly parallelizable, flexible and scalable. It can be adopted by other fields as well where an optimization analysis is required and a set of different solutions should be identified in order to perform computational experiments. Numerical examples demonstrate the applicability of the proposed strategies for realistic problems.

Appendix: Microstructure descriptors

1.1 A.1 n-point correlation function

The n-point probability functions were introduced in the context of determining the effective transport properties of random media by [2]. In Ref. [7], Debye and Buechue showed that the two-point probability functions of an isotropic porous solid can also be obtained experimentally via scattering of radiation.

The autocorrelation function of a statistically inhomogeneous system is defined as:

$$ S_2^{(j)}(\mathbf {r}_1,\mathbf {r}_2)=\langle I^{(j)}(\mathbf {r}_1)I^{(j)}(\mathbf {r}_2)\rangle $$

(6)

where r ₁ and r ₂ are two arbitrary points in the system, the angular brackets denote an ensemble average, and the characteristic function I ^(j)(r) is defined by (1). The quantity $S_{2}^{(j)}(\mathbf {r}_{1},\mathbf {r}_{2})$ can be interpreted as the probability of finding two points at positions r ₁ and r ₂ both in phase j.

For statistically homogeneous isotropic media, $S_{2}^{(j)}(\mathbf {r}_{1},\mathbf {r}_{2})$ depends only on the distance r=|r ₁−r ₂| between two points, and therefore can be expressed simply as $S_{2}^{(j)}(r)$. For all isotropic media without long-range order, $S_{2}^{(j)}(0)=\phi_{j}$ and $\lim_{r \rightarrow\infty}S_{2}^{(j)}(\mathbf {r})= \phi_{j}^{2}$ where ϕ _j is the volume fraction of phase j.

A series of significant bounds and necessary conditions on the autocorrelations (r ₁=r ₂) were derived in e.g. [17]. These provide the necessary framework for the selection of physically realizable models for the two-phase composite materials encountered in practice.

Among the significant implications of this assumption, perhaps the most interesting is the relation of the autocorrelation function with an important morphological characteristic of two-phase media called the specific surface s. In the simple one dimensional case it expresses the expectation of the number of points where a phase change occurs per unit length. It has been established that [17]:

$$ \lim_{r \rightarrow0^+} \frac{ S^{(j)}(r)-S^{(j)}(r) }{r}= -\frac{s}{2}$$

(7)

Since s must be strictly positive and finite, significant restrictions arise on the possible functional form of the autocorrelation. It is worth mentioning that similar relations can be found for two and three dimensional isotropic media as well.

The discrete nature of the digitized representation of the material allows to measure the distance r in terms of pixels or voxels and acquires integer values, with the end points of r located at the pixel (voxels) centers as shown in Fig. 21. Also, it can be shown that when sampled along the direction of rows (or columns) of pixels, S ₂(r) is a linear function between adjacent pixels:

$$S_2 (r)= (1-f)S_2(i)+f S_2(i+1) \quad\mbox{for}\ i \leq r < i+1$$

(8)

where i is an integer, and f=r mod1. Because of this linear property, the evaluation of S ₂(r) at integral values of r is sufficient to characterize the structure, and determining it for non-integer values of r is not necessary. Consequently, $S_{2}^{(j)}(r)$ can be evaluated simply by successively translating a line of r(≡i) pixels in length at a distance of one pixel at a time and spanning the whole image, counting the number of successes of the two end points falling in phase j, and finally dividing the number of successes by the total number of trials which is also the system size for a periodic medium. In 1D cases, this sampling is of course performed along the single row of pixels only.

Fig. 21

Example of discretized media and schematic representation of the two-point correlation function and the 2-point cluster correlation function. The points connected by the continuous lines fall in same cluster and contribute to both the correlation functions while the point connected by the dashed lines contribute only to the 2-point correlation function

The evaluation of the n-point correlation function of a digitized image required $O(N_{p}^{N_{p}})$ operations where N _p represents the number of pixel of the system. To date, the most efficient version of the algorithm is based on the discrete fast Fourier transform (FFT) that requires O(N _p logN _p +N _p ) operations. However, the evaluation of the correlation function due to a “local” change of the microstructure (i.e. the new configuration differs from the previous one only in n _d pixels) can be obtained in a computationally much less costly way if the correlation function “before” the change is known [27]. In practice the location of the n _d pixels are identified, computing the difference between the old and the new image, and the contribute of these pixels on the correlation function is computed, called patch correlation function (δS _n ). The correlation function is then calculated as the sum of the correlation function of the old image $S_{n}^{old}$ and the patch correlation function: $S_{n}^{new}=S_{n}^{old}+\delta S_{n}$. This method requires O(N _p n _d ) operations only.

1.2 A.2 Lineal-path

The lineal-path function L ^(j)(r ₁,r ₂) is defined as the probability of finding a line segment spanning from r ₁ to r ₂ that lies entirely in phase j [32]. This function contains some connectedness information, at least along a lineal path, and hence contains certain long-range information about the system that are very important especially for the diffusion problems. In a statistically homogeneous isotropic medium the lineal-path function depends only on the distance r between the two points and can be expressed simply as L ^(j)(r). Clearly, for all media having a volume fraction of ϕ _j :

$$ L^{(j)}(0)=S_2^{(j)}(0)=\phi_j$$

(9)

The lineal-path function can distinguish between different phases of a medium, in the sense that the lineal-path function for a particular phase does not contain connectedness information of the complementary phase(s). Therefore, for efficient reconstruction using lineal-path functions, it is important to identify which phase in the medium is the target phase to be reconstructed.

To estimate L ^(j)(r) in a digitized system one can simply sample segments of length r, and then count the number of time $N_{s}^{(j)}(r)$ that this segment following completely in the prescribed phase j. The ratio between $N_{s}^{(j)}(r)$ and the total number of trials provide an estimate of the lineal-path. However a more efficient procedure can be adopted:

• 1. introduce an oriented line into the system (Line 1 in Fig. 22);

  Fig. 22

  

  Schematic representation of the lineal-path estimation method

• 2. sample a random point (point A in the Fig. 22) belong to the phase of interest on this line;

• 3. move along the line from the sample point until a different phase is encountered (point B in the Fig. 22);

• 4. increment the counters associated with the distance r<|A−B|.

This procedure is repeated along the initial line and then repeated over many lines in a particular sample or realization. The counters are divided by the total number of the random locations chosen.

1.3 A.3 Radial distribution function

The radial distribution function, R ^(j)(r), allows to characterize dispersion of particles in very compact and efficient way. It is of fundamental importance in thermodynamics since that all the thermodynamics force can be expressed in terms of the radial distribution function. Furthermore, the radial distribution function can be ascertained from scattering experiments, which makes it a likely candidate for the reconstruction of a real system. If a priori no information is available on the nature of the material, the assumption that the material is composed by a dispersion of particles (e.g. a colloidal system) could be not adequate to describe the real microstructure of the material under analysis.

The radial distribution function represents the probability associated with finding any particle at the radial distance r from the center of another particle. To evaluate R ^(j)(r) in a digitized system, one has first to identify all the particles (or clusters) present in the medium and then to compute the center of each particle as shown in Fig. 23. It is important to remind here that the identification of the particles in a digitized medium can require a lot of computational effort. Once the centers of all the particles have been determined, the radial distribution function can be easily computed: R ^(j)(r)=N _cp (r)/N _tp where N _cp (r) represents the number of pair particles of phase j that have the distance between their centers equal to r and, N _tp represents the total number of particles in the system.

Fig. 23

Schematic representation of the radial distribution function: the lines represent the distance between different “particles”

Although the computation cost to evaluate R(r) can be quite high, in many cases this correlation function allows to reduce the complexity of the reconstruction procedure as shown in Sect. 5.3. For instance, if one is interested to reconstruct a dispersion of N _tp particles adopting the radial distribution function, the corresponding optimization problem can be reduced to identify the position of the centers of these particles and not the state of all pixels that composed the medium.