Front. Struct. Civ. Eng. 2015, 9(2): 114–120 DOI 10.1007/s11709-014-0267-5
RESEARCH ARTICLE
Simulation of heterogeneous two-phase media using random fields and level sets George STEFANOUa,b,* a
Institute of Structural Analysis & Antiseismic Research, School of Civil Engineering, National Technical University of Athens, Zografou Campus, Athens 15780, Greece b Institute of Structural Analysis & Dynamics of Structures, Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece *
Corresponding author. E-mail:
[email protected]
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2014
ABSTRACT The accurate and efficient simulation of random heterogeneous media is important in the framework of modeling and design of complex materials across multiple length scales. It is usually assumed that the morphology of a random microstructure can be described as a non-Gaussian random field that is completely defined by its multivariate distribution. A particular kind of non-Gaussian random fields with great practical importance is that of translation fields resulting from a simple memory-less transformation of an underlying Gaussian field with known second-order statistics. This paper provides a critical examination of existing random field models of heterogeneous two-phase media with emphasis on level-cut random fields which are a special case of translation fields. The case of random level sets, often used to represent the geometry of physical systems, is also examined. Two numerical examples are provided to illustrate the basic features of the different approaches. KEYWORDS
1
two-phase media, microstructure, random fields, level sets, shape recovery
Introduction
The accurate and efficient simulation of random heterogeneous media is important in the framework of modeling and design of complex materials across multiple length scales and is currently becoming one of the most active engineering research topics. In general, the stochastic differential equations governing the behavior of random multi-phase materials cannot be solved analytically. Numerical techniques are usually used for this purpose, the most prominent of which is Monte Carlo simulation (MCS) combined with the finite element method. In the framework of MCS, realizations of the random medium are generated according to its prescribed probabilistic characteristics. In various problems of engineering and applied physics, the materials under investigation consist of two distinct phases with different material properties. There are a Article history: Received May. 1, 2014; Accepted Jul. 1, 2014
number of approaches that have been proposed to reconstruct two-phase random media. The most widely used approach is based on the theory of translation fields which makes use of memory-less nonlinear transformations of Gaussian fields [1]. A special case of translation fields used to represent two-phase microstructures is that of level-cut random fields. These fields are defined by cuts of Gaussian fields above specified levels or outside bounded intervals. Alternative level-cut random field models, based on a filtered Poisson field rather than a Gaussian field, have also been proposed [2]. Another different simulation technique is based on stochastic optimization [3]. This method has the advantage of incorporating into the simulation higher order probabilistic information but its computational cost becomes excessive if a large number of realizations are required. A hybrid stochastic optimization tool, based on the synergy between various algorithms, has been recently suggested in Ref. [4] along with different computational speed-up strategies, to reconstruct a set of microstructures starting from probabilistic descriptors. A
George STEFANOU. Simulation of heterogeneous two-phase media using random fields and level sets
methodology combining the wide range of applicability of the stochastic optimization approach with the computational efficiency of the translation field approach has been proposed in Ref. [5]. This method makes use of the theory of zero-crossings of Gaussian random fields and constitutes essentially a nonlinear transformation with memory. Despite its advantages, the method is not able to capture higher order probabilistic information. In this paper, translation field models of two-phase media are examined in Section 2. The level set technique [6], originally developed for tracking moving interfaces, is presented in Section 3 as an alternative approach for the analytical definition of random (arbitrary) shapes and the reconstruction of random heterogeneous media based on available images of the microstructure. The basic features of the two approaches are illustrated through two numerical examples (Section 4). The paper closes with some concluding remarks and possible directions for future research in Section 5.
2
Random field models of two-phase media
2.1
Translation fields
It is usually assumed that the morphology of a random microstructure can be described as a non-Gaussian random field [7]. Since all the joint multi-dimensional density functions are needed to fully characterize a non-Gaussian random field, a number of studies have been focused on producing a more realistic (approximate) definition of a non-Gaussian sample function from a simple transformation of an underlying Gaussian field with known secondorder statistics. Thus, if g(x) is a homogeneous zero-mean Gaussian field with unit variance and spectral density function (SDF) Sgg(κ) (or equivalently autocorrelation function Rgg(ξ)), a homogeneous non-Gaussian stochastic field f(x) with power spectrum SfTf ðκÞ is defined as: f ðxÞ ¼ F – 1 ⋅Φ½gðxÞ,
(1)
where Φ is the standard Gaussian cumulative distribution function and F is the non-Gaussian marginal cumulative distribution function of f(x). The transform F – 1 ⋅Φ is a memory-less translation since the value of f(x) at an arbitrary point x depends on the value of g(x) at the same point only and the resulting non-Gaussian field is called a translation field [1]. This class of random fields has been used to represent various non-Gaussian phenomena, e.g., the peak dynamic response distribution of nonlinear beams or the spatial variability of the crystallographic orientation in random polycrystalline microstructures [8,9]. Translation fields have a number of useful properties such as the analytical calculation of crossing rates and extreme value distributions. They also have some limita-
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tions, the most important of which from a practical point of view is that the choice of the marginal distribution of f (x) imposes constraints to its correlation structure [10]. In other words, F and SfTf ðκÞ (or RTff ðξÞ) have to satisfy a specific compatibility condition derived directly from the definition of the autocorrelation function of the translation field: 1 1
RTff ðξÞ
¼
! ! F – 1½Φðg1ÞF – 1½Φðg2Þ
–1–1
⋅f½g1 ,g2 ;Rgg ðξÞdg1 dg2 ,
(2)
where g1 ¼ gðxÞ, g2 ¼ gðx þ ξÞ, f½g1 ,g2 ;Rgg ðξÞ denotes the joint density of fg1 ,g2 g and ξ is the space lag. If F and SfTf ðκÞ are proven to be incompatible through Eq. (2), i.e., if RTff ðξÞ has certain values lying outside a range of admissible values and/or the solution Rgg(ξ) is not positive definite and therefore not admissible as an autocorrelation function, there is no translation field with the prescribed characteristics. In this case, one has to resort to translation fields that match the target SDF approximately [11]. Alternatively, the aforementioned issue arising in the context of translation fields can be treated by using 1) an iterative procedure involving the repeated updates of the SDF of the underlying Gaussian stochastic field g(x) and, 2) an extended empirical non-Gaussian to non-Gaussian mapping leading to the generation of a non-Gaussian field f(x) with the prescribed F and SfTf ðκÞ [12,13]: f ðxÞ ¼ F – 1 ⋅F ½gðxÞ,
(3)
where F* is the empirical marginal probability distribution of g(x) updated at each iteration. The iterative updating procedure is defined in such a way that when the final realization of g(x) is generated, according to the updated Sgg(κ) and then mapped to f(x) via Eq. (3), the resulting non-Gaussian field will have both the prescribed marginal probability distribution and SDF but will not be a translation field in a rigorous sense. Sample functions of g(x) are usually generated using either the spectral representation method or the Karhunen-Loève expansion [14,15]. The extended empirical non-Gaussian to non-Gaussian mapping of Eq. (3) is used in order to overcome the possible incompatibility between the marginal distribution and the correlation structure of a translation field. Since experimental data can lead to a theoretically incompatible pair of F and SfTf ðκÞ, it is obvious that an algorithm covering a wider range of non-Gaussian fields is of great practical interest. 2.2
Level-cut random fields
Level-cut random fields are a special case of translation fields defined as follows [2]:
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( f ðxÞ ¼
1, if gðxÞ > a0 0, if gðxÞ£a0
:
(4)
In this case, the mapping of Eq. (1) has the form of the indicator function 1 (g(x) > a0) and a0 is a constant defining the level at which the cut is made. The level-cut random field of Eq. (4), where g(x) is a homogeneous Gaussian field, has been extensively used to represent twophase microstructures [2]. The sets {x2 D: f(x) = 1} and {x2 D: f(x) = 0} define the inclusions (phase 1) and matrix (phase 2), respectively. The second moment properties of a homogeneous levelcut random field are: E½f ðxÞ ¼ PðgðxÞ > a0 Þ ¼ p1 , E½f ðx1 Þf ðx2 Þ ¼Pðgðx1 Þ > a0 , gðx2 Þ > a0 Þ ¼Rf f ðx1 – x2 Þ:
(5a) (5b)
The constant p1 is equal to the volume fraction of phase 1 and the function Rff is referred to in material science as the two-point correlation (or probability) function. It is clear that the mean square is also equal to p1, thus the variance is p1(1 – p1) and Rff takes values in the range (p21 ,p1 ). It is often useful in practice to normalize the random field f(x) to have zero mean and unit variance. Level-cut Gaussian fields are in some cases inadequate to accurately describe the micro-structural features of random media because they have the limitations of the translation model mentioned in Section 2.1 [2,16]. However, in a recent paper [17], the compatibility relation between the marginal distribution and the autocorrelation of homogeneous binary valued fields has been rigorously examined and proved to be not a critical restriction for practical random media with irregular structures. In the present paper, the issue of compatibility is investigated in the numerical example of Section 4.1. Non-homogeneous (non-stationary) random fields are very often needed to describe two-phase heterogeneous random media, e.g., in the case of functionally graded materials where the microstructure is intentionally changed with location. The standard translation process theory has been extended to the non-stationary case by Ferrante et al. [18]. In that paper, it is shown that the range of admissible values of the autocorrelation function of the translation field becomes narrower in this case and therefore the issue of possible incompatibility between the marginal distribution and the correlation structure of a non-homogeneous translation field becomes more critical. The simulation of a non-stationary binary translation process, used to describe a two-phase functionally graded composite, is also presented in Ref. [18] as an illustration of the theory. It should be noted that, although the level-cut random field model is capable of producing good reconstructions in most cases, it provides only an approximation of the
original random medium based on second order probabilistic characteristics. To increase the accuracy of the reconstruction, it is necessary to incorporate other types of two-point (e.g., two-point cluster and lineal path functions) or higher order correlation functions. This is usually done in the framework of the stochastic optimization approach [3,19].
3
The level set technique
Analytically defined random level set functions such that of Eq. (11) can be directly used to produce samples of twophase media with inclusions of random arbitrary shape [20]. As the reconstruction of random heterogeneous media is often based on available images of the microstructure, the level set technique can alternatively be used to resolve the problem of shape recovery from an image [6]. The basic features of the method used for shape recovery are described below. Suppose that a contrasted image is available, defined by a mapping I : x 2 D↕ ↓R, whose value I(x) represents the grayness intensity at location x2 D. The aim is to detect the boundary of the underlying shape. This boundary is in fact located in the region where the intensity has the highest gradients. The aim of shape recovery consists in building a level set function fðxÞ whose iso-zero is located in this region. The basic idea consists in propagating a front, represented by the iso-zero of a time-dependent level set fðx,tÞ, which will “lock” on the desired boundary (Fig. 1). The equation of motion of a level set fðx,tÞ is a HamiltonJacobi partial differential equation of the form: ∂t fðx,tÞ þ vðx,tÞjjrfðx,tÞjj ¼ 0,
(6)
fðx,0Þ ¼ f0 ðxÞ, where v is the speed of the front in the outward normal direction (from negative to positive values of f). To make the iso-zero lock in high intensity gradients zones, the speed has to vanish in these zones. A classical choice for v is the following [6]: v ¼ ð1 – εÞ
1 , 1 þ cjjrðG IÞjj
(7)
where κ is the curvature of the front, ε > 0 a small parameter, I the mapping of grayness intensity and Gσ a Gaussian smoothing filter with characteristic width σ. rðG IÞ represents the gradient of the image convolved with the filter. The curvature term is a classical regularization term leading to a smooth front. Parameter c allows imposing an arbitrary small value of the speed in high intensity gradients zones. A usual choice for the initial level set f0 consists in a small circular front at the interior of the boundary to be recovered (see Fig. 1). Many algorithms have been
George STEFANOU. Simulation of heterogeneous two-phase media using random fields and level sets
proposed in order to solve the equation of motion (6) [6]. After discretization and resolution, a discretized level set f 2 RN is obtained. The technique described in this section is used in the second numerical example to reconstruct a random shape analytically defined by a random level set function. It is clarified that the Hamilton-Jacobi Eq. (6) is not needed to build the level set function, which is defined as the signed distance function to the boundary of a shape (see Eq. (11)). The Hamilton-Jacobi equation is needed to describe the evolution of the level set function in problems of shape recovery or evolving interfaces.
j j
RTff ðÞ ¼ 2f e – b ,
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(8)
where σf is the standard deviation of the random field and b is its correlation length. The corresponding SDF is: SfTf ðÞ ¼
2f d , π d2 þ 2
(9)
with d = 1/b. In this example, it is assumed that σf = 0.5, b = 5 and the volume fraction of the level-cut random field p1 = 0.5. The underlying Gaussian field is simulated using the spectral representation method which has the advantage of producing ergodic sample functions in a sample by sample sense [14]. It is worth mentioning that, in the special case of p1 = 0.5, the solution of Eq. (2) leads to an analytical relationship between the two correlation functions [22]: π Rgg ¼ sin RTff : (10) 2 From Eqs. (8) and (10), it follows that both correlation functions take values in the range (0,1] and thus RTff should be compatible with the marginal distribution of the levelcut random field according to translation field theory. Sample functions of the reconstructed random medium are generated using the algorithm of Ref. [12]. (Fig. 2). Equation (10) shows that the two correlation functions have substantial differences and thus the algorithm needs a large number of iterations to converge if Sgg(κ) is initially set equal to the target spectrum SfTf ðÞ. The spectral density function of the simulated level-cut field is shown in Fig. 3. A good agreement can be observed between the target and approximated SDF of f(x).
Fig. 1 Shape recovery using the level set technique: A small circular front propagates until it “locks” on the desired shape
4
Fig. 2 Sample realization of random medium with the two-point correlation function of Eq. (8)
Numerical examples 4.2
2-D random media with inclusions of arbitrary shape
4.1 1-D Debye random medium
A one-dimensional random medium is simulated in this example using the theory presented in Section 2. According to Debye [21], microstructures in which one phase consists of random shapes and sizes can be described by the exponentially decaying two-point correlation function
In this example, samples of a random shape are obtained using the shape recovery technique based on the level set method described in Section 3 of the paper. The shape to reconstruct is analytically defined by a random level set function in a square domain [0,1] [0,1] as follows [23]: fðx,Þ ¼ jjx – cjj – Rða,Þ:
(11)
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Fig. 3 SDF of sample function shown in Fig. 2 versus target SDF
The iso-zero of level set f is a random “rough” circle C(θ) shown in Fig. 4 (θ denotes the randomness of a quantity). c is the “center” of the rough circle and R(α,θ) is a random field representing its radius indexed by the angle a of the polar coordinate system centered at c, defined as: Fig. 5 Gradient intensity of three filtered sample images (thick black line) and corresponding iso-zero of the recovered level sets (thin blue line)
Rða,Þ ¼ 0:2 þ 0:03Y1 ðÞ þ 0:015½Y2 ðÞcosðk1 aÞ þ Y3 ðÞsin ðk1 aÞ þ Y4 ðÞcosðk2 aÞ þ Y5 ðÞsin ðk2 aÞ,
(12)
where k1, k2 are deterministic constants and Y1 ðÞ,:::,Y5 ðÞ are independent identically pffiffiffi pffiffiffi distributed uniform random variables Yi 2 U ð – 3, 3Þ, i = 1,…,5. With an appropriate choice of the speed of the front in Eq. (7) where ε = 0.091, the initial curvature κ = 10 and c = 1, the shape recovery technique is applied n times to produce n samples of the random rough circle. It is worth noting that, since the recovery procedure involves calculation of image gradients whose accuracy depends on the mesh size, a sufficiently fine mesh must be used in order to be able to capture details of shape features. In this case, a 100 100 mesh has been used to this purpose. Figure 5 illustrates the filtered gradients of three sample images I(k) and the corresponding iso-zero of the recovered
Fig. 4 Schematic representation of a “rough” circle
level sets. The sample images are “manufactured” using Eqs. (11), (12) with parameters (k1, k2) equal to (0,3), (0,6) and (2,6), respectively. Figure 6 shows the initial and recovered level sets for the three cases considered. It can be observed that an accurate reconstruction of the random shape is obtained using the level set-based recovery technique. Due to the moving interface nature of the level set method, the approach can be particularly useful in simulating the morphological evolution of microstructure in time such as in binary alloys [24]. The level set technique is also often used in the computational homogenization of random heterogeneous media where a
Fig. 6 Initial (in magenta) and recovered level sets (in blue) for (k1, k2) = (0,3), (0,6) and (2,6)
George STEFANOU. Simulation of heterogeneous two-phase media using random fields and level sets
large number of samples of microstructures with multiple inclusions are needed [25]. Recent examples can be found in [20,26] where the homogenization of two-phase random media is performed in the framework of the extended finite element method – XFEM (Fig. 7).
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Lifelong Learning” (Action’s Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State. The provided financial support is gratefully acknowledged by the author. Special thanks are also due to Professor George Deodatis for fruitful discussions on the topic of simulation of heterogeneous media.
References
Fig. 7 Sample realization of random microstructure with inclusions of irregular shape generated using Eqs. (11) and (12) with (k1, k2) = (0,6)
5
Conclusions
In this paper, the simulation of heterogeneous two-phase media has been examined with emphasis on level-cut random fields, which are a special case of translation fields, and on the level set technique. Level-cut Gaussian fields are capable of reproducing accurately a broad range of two-phase microstructures. From a practical point of view, their most important limitation is the possible incompatibility between their marginal distribution and two-point correlation function. However, recent research has shown that the issue of compatibility is not a critical restriction for many practical random media with irregular structures. Level sets offer an accurate reconstruction of random geometry and, in the context of heterogeneous media, can be used for reproducing inclusions of various shapes as well as for shape recovery from available images of the microstructure. The incorporation of higher-order probabilistic information into the available random field models and the representation of statistically inhomogeneous media (e.g., functionally graded materials) by a possible combination of the existing approaches constitute important future research directions. Acknowledgements This work was implemented within the framework of the research project “MICROLINK: Linking micromechanics-based properties with the stochastic finite element method: a challenge for multiscale modeling of heterogeneous materials and structures” - Action “Supporting Postdoctoral Researchers” of the Operational Program “Education and
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