Wavelet-based analysis of multiscale phenomena: application to material

porosity and identification of dominant scales

G. Frantziskonis*

Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721-0072, USA

Abstract

The paper presents a general process that utilizes wavelet analysis in order to link information on material properties at several scales. In

the particular application addressed analytically and numerically, multiscale porosity is the source of material structure or heterogeneity, and

the wavelet-based analysis of multiscale information shows clearly its role on properties such as resistance to mechanical failure.

Furthermore, through the statistical properties of the heterogeneity at a hierarchy of scales, the process clearly identifies a dominant scale or

range of scales. Special attention is paid to porosity appearing at two distinct scales far apart from each other since this demonstrates the

process in a lucid fashion. Finally, the paper suggests ways to extend the process to general multiscale phenomena, including time

scaling. q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Multiscale; Porosity; Wavelets; Compound matrix

1. Introduction

The relationship between material microstructure and

material properties such as mechanical, electrical, etc.

continues to be an active research field in the engineering,

materials science, and physics communities. Each group

concentrates, in general, on the microstructure at specific

(spatial and temporal) scales; as a result, there is practically

little communication between groups studying similar

problems, yet at different scales. Even though research

from diverse groups have resulted to a better understanding

of issues such as deformation, failure, toughness, electrical

conductance, it has been recognized that several open

questions in this field can greatly benefit from the

recognition that material heterogeneity (microstructure)

depends on scale, i.e. it is multiscale for (most) materials.

Even though information at several scales can be obtained,

e.g. by observing material properties at a range of scales or

by performing simulations at several scales, from atomic to

macro, [1], it is not clear how to link such information along

scales. As an example, consider the case, where porosity is

the source of heterogeneity and it realizes itself differently at

various scales. For illustration purposes, consider a two-

scale heterogeneity; at a scale large enough only the spatial

distribution of ‘large’ pores is observable, while at small

scales only that of the ‘small’ pores can be seen. Even if the

structure of the pores at these two distinct cases is fully

specified or observed, it is difficult to compare the role of the

porosity at each of the two scales on overall material

properties. In other words, for, say, mechanical or electrical

breakdown (failure), it is difficult to decide whether the

large or the small pores are of most importance.

Examples as the porous material above or materials with

multiscale microstructure are rather ubiquitous. For

example, the notion of order/disorder in micro/macroscopic

physics is scale dependent; ceramics may show various

porosity distributions at scales ranging from nanometers to

millimeters. As a more general example of multiscale

material structure, consider polycrystalline materials in

which point defects, dislocations, grain boundaries, grain

boundary junctions, precipitates and pores span a very wide

range of scales from the atomic to the mesoscopic to the

macroscopic. It is difficult to compare the role of each of

these microstructures on the properties of the material.

In more general terms, heterogeneity manifests itself on a

hierarchy of scales and this brings about the importance of

multiscale analysis tools; this comprises the central theme

of the present work. Furthermore, the dynamic nature of

several related phenomena (e.g. as it pertains to the

relaxation time of different microstructures) brings about

time scaling in addition to the spatial one; even though the

present paper does not address time scaling, it is feasible to

incorporate it.

0266-8920/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 26 6 -8 92 0 (0 2) 00 0 32 -2

Probabilistic Engineering Mechanics 17 (2002) 349–357

www.elsevier.com/locate/probengmech

* Tel./fax: þ1-520-621-4347.

E-mail address: frantzis@email.arizona.edu (G. Frantziskonis).

Multiscale phenomena have recently received much

attention in several branches of physical science. In

materials, a large part of the work is devoted to modern

simulation methods and coupling of length scales; they can

be characterized as either serial or concurrent. In serial

methods a set of calculations at a fundamental level (small

length scales) is used to evaluate parameters for use in a

more phenomenological model to describe the phenomenon

of interest at longer length scales. For example, atomistic

simulations can be used to feed the constitutive behavior of

finite elements, which are then used to simulate large-scale

problems [2]. Concurrent methods rely on different

computational methodologies applied to different regions

of a material. For example, the problem of crack

propagation is a problem that was tackled early on (and

still is) by atomic simulations techniques near the tip of a

crack, where large deformations (even bond breakage)

occur and continuum approaches at large distances from the

tip of the crack. This has been addressed by dividing space

into two regions: namely the tip of the crack, where the

material is treated atomistically with molecular dynamics

and the rest of the material modeled as a continuum via the

finite element method [3,4].

The compound wavelet matrix (CWM) method intro-

duced in Ref. [5] consists of a different approach in which

various computational methodologies (the process is

general, yet molecular dynamics simulation and Monte

Carlo simulation of Q-states Potts model have been

illustrated in Ref. [5]) are applied to a region of material

simultaneously at the coarse and the fine spatial scales.

Matrices of the wavelet coefficient are produced from

energy maps representing the spatial distribution of the local

excess energy in the microstructures obtained with both

methodologies. The full description of the material is then

obtained by merging the matrices of the wavelet coefficients

representing the material at different scales through the

CWM. The CWM then characterizes the material over a

range of scales that is the union of the scales treated by the

two methodologies.

The present paper introduces a general framework for

comparing information obtained (through simulations, or

experimentally or through combined simulation–exper-

imental procedures) at several scales. It complements the

CWM method since it deals with identifying specific

measures for comparison of information at various scales

(CWM does not do that). Furthermore, it is applicable to

general type of available data, e.g. from experiments,

simulations, etc. Even though the process is detailed with

respect to multiscale porosity, it can, in general, be extended

to various forms of material heterogeneity. It will become

apparent that the process has to be adapted appropriately to

the physical problem at hand; yet the ‘backbone’ of the

process remains essentially the same. Finally, the process is

general enough and can be extended to problems with

temporal scaling as well. The rest of the paper is organized

as follows. Section 2 provides relevant ‘background’ and is

followed by the wavelet analysis of porous media and

implications on properties such as mechanical failure

(Section 3). The wavelet-based study of multiscale

phenomena is presented in Section 4, which examines the

two-scale porosity case as a core one, with respect to the

example application of mechanical failure. Appendix A

provides mathematical details on some properties of the

wavelet analysis of porous media, and the conclusions

(Section 5) address some possible extensions, including

time scaling.

2. General properties of porous media

Most of the material in this section is background,

included for convenient subsequent referencing and for

making the paper as self-contained as possible; it can be

traced from several text books, [6] (see also references in the

sequence). Let us consider a general porous material, where

the matrix is assumed homogeneous and no restrictions are

imposed on the form of pores (also termed flaws in the

sequence). The geometry of the porous medium can be

described with the aid of the fundamental function which is

defined as being unity for spatial locations in the matrix and

as zero for locations in the flaws. The specific area of flaws

is defined as

s ¼ limDV!0

kDAlDV

ð1Þ

where DV denotes an infinitesimal volume and DA is the

portion of the solid–flaw interface crossing DV. It is thus the

ensemble average, k·l, of the density of solid–flaw interface

elements. The specific area s is related to the autocorrelation

function of the solid phase rðrÞ; where for statistically

stationary media, r is the vector designating the spatial

distance between two points and for stationary as well as

isotropic media r is merely the magnitude of that vector. The

following holds for stationary and isotropic media [7,8]

s ¼ 24qð1 2 qÞ›rðrÞ

›r

����r¼0

ð2Þ

where q denotes porosity, i.e. volume of flaws over total

volume. The correlation distance of the solid phase, d, is

expressed as

d ¼4qð1 2 qÞ

sð3Þ

and the variance of the solid phase is given by

s2f ¼ qð1 2 qÞ ð4Þ

The autocorrelation function is exponential, i.e. Ref. [6] and

references therein

rðrÞ ¼ expð2r=dÞ ð5Þ

with the correlation distance d expressed in Eq. (3). For

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357350

quite general cases of flaw spatial arrangements [6]

s ¼q

c1

ð6Þ

where c1 is dictated by the arrangement of flaws; as

examples, for uniform spheres of radius R, c1 ¼ R=3; for

lognormal distribution of flaw sizes c1 is proportional to the

mean flaw size over the square of the coefficient of

variation. For such cases, 4qð1 2 qÞ=d ¼ q=c1; thus

s2f < q

d

4c1

ð7Þ

From the above, the total interface area At is expressed as

At ¼ sV ¼qV

c1

ð8Þ

Using Eqs. (3) and (6) it follows that

d ¼ 4ð1 2 qÞc1 ð9Þ

3. Wavelet transform of porous media and relevant

properties

The wavelet transform can be seen as a mathematical

microscope: increasing magnification provides insight into

the intricate structure of a ‘pattern’. Wavelets provide

means to mathematically represent functions space and/or

time- and scale-wise with a few parameters. The so-called

wavelet coefficients provide local information on the

function and also information relevant to scale (level of

magnification in space/time). Fourier transforms based on

combinations of sines and cosines are very useful for

describing periodic and stationary functions, but they are

not useful for the many non-periodic, non-stationary

phenomena encountered in the physical world. Furthermore,

wavelet coefficients provide local information, important

for several problems. Use of wavelets has found riches in

analyzing complex signals and structures, and its use as a

tool for connecting information at various scales (multiscale

phenomena) is rather at formative stage; some recent

attempts can be found in Ref. [9].

A more general goal of wavelet analysis is to describe

graphs (signals) as superposition of elementary functions.

The corresponding description is then used for different

purposes, i.e. data compression, feature extraction, pattern

recognition, etc. There are several publications on this

rather new subject and applications can be found in a wide

variety of scientific/engineering fields; over the past few

years, several dozens of monographs have appeared in the

literature. Wavelet transforms provide both scale and

location (and/or time) information about a given function.

A wavelet cðxÞ transforms a function f ðxÞ according to

Wfða; bÞ ¼ðRD

f ðxÞca;bðxÞdx ð10Þ

The two-parameter family of functions

ca;bðxÞ ¼1

aD=2c

x 2 b

a

� �ð11Þ

is obtained from a single one, c, called the mother wavelet,

through dilatations by the factor a and translations by the

factor b. Here, D denotes the spatial dimensionality of the

problem and a [ Rþ; a – 0; and b; x [ RD: The factor

1=aD=2 is included for normalization purposes and, with it all

the wavelets have the same energy. Other types of

normalization, however, are possible. The scalars defined

in Eq. (10) measure the fluctuations of f ðxÞ around point b,

at scale a.

A wavelet analysis can either be continuous or discrete.

The second one, based on an orthogonal decomposition of a

signal can be performed with fast algorithms. Given the

wavelet transform Wfða; bÞ associated to a function f, it is

possible to reconstruct f and/or construct its representation

at a range of scales between s1 and s2 ðs1 # s2Þ through the

inversion formula

fs1;s2ðxÞ ¼

1

cc

ðs2

s1

ðRD

Wfða; bÞca;bðxÞdbda

a1þDð12Þ

By setting s1 ! 0; s2 !1; f is reconstructed.

The variance, k½Wfða; bÞ�2l provides means for evaluating

the energy of the wavelet transform as a function of scale a.

It can be evaluated in the Fourier space as

s2WðaÞ ¼ k½Wfða; bÞ�

2l ¼aD

2p

ðRD

PfðkÞ½cðakÞ�2dk ð13Þ

where Pf ðkÞ denotes the power spectrum of f and ð·Þ

indicates Fourier transform. Consider the energy of the

wavelet transform at a scale a and at a specific spatial point

b. Here, the local energy, considering normalization

expressed in Eq. (11), is defined as

ElðaÞ ¼ðWfða; bÞÞ

2

a1þDð14Þ

This energy, for small a (a ! 0) is non-zero only in the

neighborhood of the solid–flaw interfaces. (The scaling of

such energy, for small a, can be evaluated from the local

regularity of the function, but such an approach is out of the

scope of the present work; some relevant issues are

addressed when appropriate, however). Then, for the

isotropic media considered herein, the total energy of the

wavelet transform at scale a is simply the product of

the local energy multiplied by At. Thus, for scale a

sufficiently small, if the local energy is designated as El;

the normalized (over the domain) energy of the wavelet

transform, EW ðaÞ, is expressed using Eq. (8) for a single

realization of the porous medium, as

EWðaÞ <AtEl

V¼

q

c1

ElðaÞ ð15Þ

In the following, unless indicated otherwise, we define EW

to be the energy of the transform at the first level of the

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357 351

(discrete) wavelet decomposition, i.e. at the smallest scale a,

i.e. EW ¼ EWðaÞlMinðaÞ:

From Eq. (15) it follows that

q ¼c1EW

El

ð16Þ

where El and EW are evaluated at the minimum scale a. Let

us consider the local energy of the wavelet transform of the

medium with flaws. We consider a discrete transform and

the minimum scale addressed above is the first scale of

decomposition. At a solid–flaw interface the medium is

represented mathematically through the fundamental func-

tion, which is realized as a local jump at the finest scale,

provided the discretization is fine enough relative to the size

of the pores. Under these conditions, then, due to the

constant size of the jump resulting from the fundamental

function defining the porous medium, the local energy of the

wavelet transform at the finest scale is constant at the solid–

flaw interfaces. Even though this justification is enough, in

passing, and without resorting to a rigorous proof, it is noted

that under such conditions of discretization, the porous

medium can be approximated as being C 0 continuous with a

singularity exponent [10] approaching zero, and thus for fine

scales (a ! 0) the wavelet transform is ,O[a 0], i.e.

constant [11]. Then, with respect to the minimum scale a,

under the conditions for discretization discussed earlier, at

solid–flaw interfaces

El < const: ð17Þ

Energy EW is expressed as, assuming ergodicity

EWðaÞ <

1

a1þD

ðVðWfða; bÞÞ

2db

V¼

1

a1þDs2

WðaÞ ð18Þ

In Appendix A we provide certain relevant examples on the

scaling of the energy of the wavelet transform, where it is

shown that for small a, s2W; expressed through Eq. (13)

scales as s2f a1þDd21: Then

EW <s2

f

dð19Þ

The same type of relation as Eq. (19) results from Eqs. (15),

(17) and (7); this, then implies that the relation crucial to this

work, i.e. Eq. (16), is valid for general porous media, or,

equivalently for media with exponential spatial correlation

structure. Even though Appendix A considered only one

wavelet expression, in the following it will be shown

numerically that the conclusions of Appendix A hold also

for other wavelets, for media possessing exponential

autocorrelation. Then, this suggests that Eq. (16) is

independent of the wavelet used (even though only few

wavelets were used in the numerical results).

Eq. (19) can be examined numerically by creating

random fields with exponential autocorrelation and speci-

fied s2f and d. As is well known, different physical

procedures can lead to the same autocorrelation structure

[12]. For detailed, enlightening, and justifying discussion of

the correlation structure of fields as it may pertain to

different physical procedures, e.g. an exponential auto-

correlation pertaining to a porous medium, we refer to Ref.

[12]; here, we find appropriate and relevant to the results

presented subsequently to quote part of that book, p. 50, that

‘…for any random function XðtÞ with finite first and second

moments it is always possible to construct a Gaussian

function X0ðtÞ with the same mean value and the same

correlation function as XðtÞ’. Thus, Eq. (19) has been

examined numerically, by creating random fields with

exponential autocorrelation using the spectral represen-

tation method of Ref. [13]. EW shown in Fig. 1 was

calculated from 10 realizations of 256 £ 256 fields, where a

value of sf ¼ 0:1 was used and d varied accordingly.

Similar results were obtained in one-dimension using 400

realizations of 256-point processes. In passing it is noted

that in both one and two dimensions the value of EW

converged for less simulations than actually used; the small

number of simulations needed and the relatively fast

algorithm for creating the fields makes a detailed conver-

gence study rather unnecessary at this stage. The wavelet

used for calculating EW was a biorthogonal spline with four

vanishing moments [14]; a non-systematic study showed no

influence of the wavelet on the results. Fig. 2 was produced

similar to Fig. 1, and, as can be seen, for the Gaussian

Fig. 1. Ratio s2f =d versus EW for exponential autocorrelation. Fig. 2. Ratio s2

f =d versus EW for Gaussian autocorrelation.

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357352

autocorrelation (different than the exponential which is

valid for the porous media) the linear trend does not hold.

Figs. 3 and 4 correspond to Figs. 1 and 2, respectively,

and show the variation of EW as a function of d for constant

sf. For clarity, the values of d in Fig. 3 are 5/8, 5/4, 5/2,…,

20, 40 units, while in Fig. 4 they are 5/16, 5/8, 5/16;…; 5; 10

units.

Let us examine the implications of the porosity as

described above on a representative property, i.e. mechan-

ical failure; other properties can, however, be studied in a

similar fashion. From Refs. [15–17], the ratio of failure

macroscopic (nominal) stress sc over the failure stress of the

background homogeneous system (matrix material without

the pores), s0 is expressed as

sc

s0

<1

1 þ k 2ln V

ln q

� �a� � ð20Þ

where 1=½2ðD 2 1Þ� # a # 1 and q denotes the volume

fraction of voids. The exponent a assumes its lower value

for an isolated flaw while its upper value considers

interaction of flaws. Eq. (20) holds for q far from its

percolation threshold value; yet, near percolation, a similar

relation holds as shown in Ref. [15]. Eq. (20) was derived

from the fact that in the porous medium the typical size of

the largest crack-like flaw S behaves as S < 2ln V=ln q:

Furthermore, it accounts for the fact that the ratio of

maximum local stress smax over s0 scales as 1 þ KS1=2 for

an isolated flaw and as 1 þ KS when flaw interactions are

included. From Eqs. (7), (16), and (20), setting c ¼ 4c1; we

obtain

sc

s0

<1

1 þ k 2ln V

lncs2

f

d

!0BBBB@

1CCCCAa266664377775

¼1

1 þ k 2ln V

lnc1EW

El

� �0BB@

1CCAa26643775

ð21Þ

Eqs. (16) and (20) hold for any volume V as far as the flaw

space in it is statistically stationary. Eq. (21) is used in

Section 4, where multiscale analysis of the phenomenon is

addressed.

4. Multiscale analysis

In order to compare information at various scales, let us

consider, without loss of generality, the case of two distinct

scales far apart from each other; this demonstrates the

process clearly, and its extension to more general cases is

straightforward. The problem addressed through Eq. (20) or

(21), is to identify the contribution from the microstructure

(porosity in our case) as it appears at each scale. A way to do

this is through the wavelet transform of the entire porous

medium, using appropriate spatial correlation functions.

The medium with two distinct scales can be considered as a

‘composite’ one [18] with statistical properties directly

dependent on those at the two distinct scales. Let d1, d2 be

the correlation distance for the large-scale, small-scale

porosity structure, respectively, and d1 q d2: Also, let s2f1;

s2f2

be the corresponding variances, related to porosities q1,

q2 through Eq. (4). The composite medium has the

following properties [18]; autocorrelation function

rðrÞ ¼ l1r1ðrÞ þ l2r2ðrÞ ð22Þ

autocorrelation distance

l ¼ l1d1 þ l2d2 ð23Þ

and variance

s2f ¼ s2

f1þ s2

f2ð24Þ

where for the large-scale process the autocorrelation

function is r1ðrÞ ¼ exp½2r=d1�; variance is s2f1

and for the

small-scale process r2ðrÞ ¼ exp½2r=d2� and variance is s2f2:

Finally, l1 ¼ s2f1=ðs2

f1þ s2

f2Þ and l2 ¼ s2

f2=ðs2

f1þ s2

f2Þ: The

three relations Eqs. (22)–(24) can be used in conjunction

with Eqs. (20) and (21) to study fracture of porous materials

Fig. 3. Correlation distance d versus EW for exponential autocorrelation and

best fit by a function ,1/x.

Fig. 4. Correlation distance d versus EW for Gaussian autocorrelation and

best fit by a function ,1/x.

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357 353

with two-scale porosity. For this purpose random fields with

autocorrelation Eq. (22) are produced since the spectral

density function of such a process is, in two dimensions,

where k ¼ ðk21 þ k2

2Þ21=2

PfðkÞ ¼s2

f1

2pd1

1

d21

þ k2

!3=2þ

s2f2

2pd2

1

d22

þ k2

!3=2ð25Þ

In order to identify the role of microstructure (porosity in

the present case) at each scale, the central theme of this

paper, the variation of EW as a function of scale should be

known. For this, it is appropriate to decompose information

scale-wise and since this decomposition is linked together

through the wavelet basis we can study the contribution of

each scale. Thus, since q is the basic variable that enters the

mechanical failure criterion, we rewrite Eq. (4) in the

following form, also using Eq. (19)

q ¼ ð1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 4s2

f ðaÞq

Þ=2 < s2f ðaÞ < dEWðaÞ ð26Þ

and the first < holds for q close to zero and far from the

percolation threshold. In Eq. (26) s2f ðaÞ; EWðaÞ denote the

variance, energy of wavelet transform at scale a, respect-

ively. These quantities can be evaluated from the wavelet

decomposition of the field at each scale—note that each

decomposition scale is the first detail scale from the

previous one—or from the wavelet representation of the

field at each scale, i.e. through Eq. (12). Let us consider

the case, where the domain is 256 £ 256 units, d1 ¼ 50

units, d2 ¼ 0:3 units, sf1¼ 0:1 ðq1 ¼ 0:0101Þ and sf2

varies

between 0.1 ðq2 ¼ 0:0101Þ and 0.4 ðq2 ¼ 0:2Þ: The follow-

ing results are from 10 realizations of the relevant

256 £ 256 fields. Of course, larger processes can be

simulated, yet, what was found, as expected for the

problems considered herein, as the domain size increases

the number of realizations needed for convergence

decreases. The number of realizations for convergence in

our case, with the large correlation distance being equal to

50 units and the domain size being 256 £ 256 units, was less

than 10; yet we used 10 realizations for all the results shown

in the sequence. Fig. 5 shows an animation of Eq. (26),

where the horizontal axis denotes scale a and the vertical

wavelet representation of q or dEWðaÞ evaluated from 10

realizations of the process and evaluation of Eq. (26).

Clearly, these plots show two local peaks, at the scale of the

large and small correlation distances of 50 and 0.3 units,

Fig. 5. Animation of Eq. (26) for sf2equal to 0.1, 0.125, 0:15;…; 0:4 (q ¼ 0.0101, 0.0158, 0:023;…; 0:2) from left to right and from top to bottom. The

horizontal axis in each plot denotes scale. Scale 0 (zero) corresponds to 1 point in the wavelet decomposition, i.e. the coarsest available scale (256 units) scale 1

to 256=2 ¼ 128; scale 3 to 64 units,…, scale 8 to the finest detail scale, i.e. 2 £ 2 units of the 256 £ 256 domain; scale 8 also corresponds to the first wavelet

decomposition, scale 7 to the second,…,etc.

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357354

respectively. For small q2 clearly the large-scale porosity is

dominant and will dominate the fracture process. However,

as the small-scale porosity increases, the importance

(contribution) of the small-scale fluctuations increase and

for high enough q2 its contribution surpasses that of the

large-scale ones. As can be seen from Fig. 5, for sf2equal to

about 0.25 (q2 < 0.07) the contribution from the small-scale

porosity is the same as that of the large-scale one. It is

interesting to study the conditions under which this happens,

i.e. equal contribution from the two scales; this can be

accomplished numerically. For the sake of illustration we

‘fix’ the correlation distances at d1 ¼ 50 and d2 ¼ 1 for the

domain size at 256 £ 256 units and vary the porosities until

the equal energy criterion is met. Table 1 shows such values

and Fig. 6 shows the corresponding plot.

5. Conclusions

A general process for linking information at several

scales is advanced and examined in detail with respect to

statistically stationary and isotropic porous media. The

example application is mechanical failure; yet, the process

is general enough and can be extended to any properties,

where scale hierarchy is important. Identification of a

dominant scale or range of scales is a natural consequence

of the multiscale description and a relevant illustration

addresses two-scale porosity.

Acknowledgements

Support from the National Science Foundation under

grant no. CMS-9812834 is gratefully acknowledged.

Appendix A

Let us first consider a process f in one dimension, i.e.

D ¼ 1; and then a field in higher dimensions. The

autocorrelation is of the exponential form

rðrÞ ¼ exp½2r=d�

and the variance is s2f : The Fourier transform of this

autocorrelation function together with s2f provide the power

spectrum of the process (Wiener–Khinchin theorem), i.e.

PfðkÞ ¼1

p

s2f d

1 þ d2k2ðA1Þ

Let us consider the so-called ‘Mexican hat’ wavelet or the

second derivative of the Gaussian function, and address the

generality of the process in the sequence, i.e.

cðxÞ ¼ ðx2 2 1Þexp½2x2=2� ðA2Þ

Its Fourier transform is

cðkÞ ¼ k2 exp½2k� ðA3Þ

Integral (13) for the functions Eqs. (A1) and (A3) yield, for

D ¼ 1

s2WðaÞ ¼ k½Wfða; bÞ�

2l ¼a1

2p

ðRl

PfðkÞ½cðakÞ�2 dk

¼as2

f

2ffiffip

pd5

ðad2ðd2 2 2a2Þ þ 2a4dffiffip

p

exp½a2=d2�ð1 2 erf½a=d�ÞÞ ðA4Þ

where erf[·] denotes the error function. The derivations and

some of those in the sequence were performed using the

program Mathematica. Expanding the last expression in

Eq. (A4) into a Taylor series around a ¼ 0 yields

s2WðaÞ ¼

s2f a2

2ffiffip

pdþ O½a�4 ðA5Þ

However, if a different autocorrelation function is used, e.g.

of Gaussian type

rðrÞ ¼ exp½2pr2=4d2�

with variance s2f and power spectrum

PfðkÞ ¼1

ps2

f d exp½2k2d2=p� ðA6Þ

using the wavelet expressed through Eqs. (A2) and (A3), it

Table 1

Values of sf1ðq1Þ and sf2

ðq2Þ so that there is equal contribution from the

small and large scales

d1, d2 sf1ðq1Þ sf2

ðq2Þ

50, 1 0.2 (0.0417) 0.165 (0.028)

50, 1 0.1 (0.0101) 0.082 (0.0067)

50, 1 0.27 (0.0792) 0.23 (0.056)

50, 1 0.3 (0.1) 0.26 (0.073)

Fig. 6. Plot of the values of q1 versus q2, shown in Table 1 and best linear fit

(solid line).

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357 355

follows that

s2WðaÞ ¼ k½Wfða; bÞ�

2l ¼a1

2p

ðRD

PfðkÞ½cðakÞ�2 dk

¼3p2s2

f

4

a5d

ðpa2 þ d2Þ5=2ðA7Þ

Expanding the last expression in Eq. (A6) into a Taylor

series around a ¼ 0 yields

s2WðaÞ ¼ O½a�5 ðA8Þ

Thus, s2WðaÞ scales as s2

f alþ1d21 for the exponential

autocorrelation function but not for the Gaussian one. In

this paper we examine exponential autocorrelation struc-

ture, resulting from the porous material.

In higher dimensions, similar results can be obtained, i.e.

for the case, where the relevant field as well as the wavelet

analysis is isotropic (spherically symmetric). In this case,

since the Fourier transform preserves the spherical sym-

metry, the power spectrum is a function of the magnitude of

the D-dimensional vector ekk; i.e. function of k ¼ ½k21 þ k2

2 þ

· · · þ k2D�

1=D: For the exponential D-dimensional isotropic

autocorrelation the power spectrum reads [12]

PfðkÞ ¼G ½ðD þ 1Þ=2�

pðDþ1Þ=2

s2f

dð1=d2 þ k2ÞðDþ1Þ=2ðA9Þ

where G [·] denotes the gamma function, and for the

Gaussian autocorrelation

PfðkÞ ¼1

pDs2

f dD exp½2k2d2=p� ðA10Þ

The so-called radial spectral density function RfðkÞ [18]

satisfies

s2f ¼

ð1

0RfðkÞdk ðA11Þ

while

s2f ¼

ðRþ

D

Pfð~kÞ d

~k ðA12Þ

and [18, p. 105]

RfðkÞ ¼pD=2

2D21G½D=2�kD21PfðkÞ; D ¼ 1; 2;… ðA13Þ

Following a process similar to that for D ¼ 1, we obtain

s2WðaÞ ¼ k½Wfða; bÞ�

2l ¼aD

2p

ðRD

PfðkÞ½cðakÞ�2 dk

¼p2s2

f

2D

aDþ4dD

ðpa2 þ d2Þ2þD=2

G ½2 þ D=2�

G ½D=2�ðA14Þ

for the Gaussian correlation, and

s2WðaÞ ¼ k½Wfða; bÞ�

2l ¼aD

2p

ðRD

PfðkÞ½cðakÞ�2 dk

¼a1þDs2

f

3d421þDG ½D=2�ð8a3G ½2 þ D=2�

£F½2 þ D=2; 5=2; a=d� þ 3d3G ½ð1 þ DÞ=2�

£F½ð1 þ nÞ=2;21=2; a=d�Þ

ðA15Þ

for the exponential autocorrelation, where F½A;B;C� is the

Kummer hypergeometric function. Similar to the D ¼ 1

case, a series expansion of Eq. (A14) at a ¼ 0 yields

s2WðaÞ ¼ O½a�4þD ðA16Þ

while Eq. (A15) yields

s2WðaÞ ¼

s2f a1þD

2ffiffip

pd

þ O½a�3þD ðA17Þ

Even though this scaling was analytically shown to hold for

a specific wavelet, numerical simulations presented in this

paper indicate it also holds for the discrete wavelets used

numerically. It may be interesting to examine whether this

scaling holds only for the exponential autocorrelation,

though this is rather irrelevant to the present study.

References

[1] Olson GB. Computational design of hierarchically structured

materials. Science 1997;277:1237–41.

[2] Tadmor EB, Phillips R, Ortiz M. Mixed atomistic and continuum

models of deformation in solids. Langmuir 1996;12:4529–34.

[3] Mullins M, Dokainish MA. Simulation of the (001) plane crack in a-

iron employing a new boundary scheme. Phil Mag A 1982;46:771.

[4] Kitagawa H, Nakatami A, Sibutani Y. Molecular dynamics study of

crack process associated with dislocation nucleated at the tip. Mater

Sci Engng 1994;A176:263–75.

[5] Frantziskonis G, Deymier PA. Wavelet methods for analyzing and

bridging simulations at complementary scales—the compound

wavelet matrix and application to microstructure evolution. Model-

ling Simul Mater Sci Engng 2000;8:649–64.

[6] Dagan G. Flow and transport through porous formations. Berlin:

Springer; 1989.

[7] Debye P, Anderson HR, Brumberger H. Scattering by an Inhomo-

geneous solid. II. The correlation function and its application. J Appl

Phys 1957;28:679–83.

[8] Matheron G. Elements pour Une Theorie des Milieux Poreux. Paris:

Masso et Cie; 1967.

[9] Fang LZ, Thews RL, editors. Wavelets in physics. Singapore: World

Scientific; 1998.

[10] Mallat S, Hwang W-L. Singularity detection and processing with

wavelets. IEEE Trans Inform Theor 1992;38:617–43.

[11] Carmona R, Hwang W-L, Torresani B. Practical time–frequency

analysis. San Diego: Academic Press; 1998.

[12] Yaglom AM. Correlation theory of stationary and related random

functions I,II. New York: Springer; 1987.

[13] Shinozuka M, Deodatis G. Simulation of stochastic processes by

spectral representation. Appl Mech Rev 1991;44:191–203.

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357356

[14] Daubechies I. Ten lectures on wavelets. Philadelphia, PA: SIAM;

1992.

[15] Duxbury PM. In: Herrmann HJ, Roux S, editors. Statistical models for

the fracture of disordered media. Amsterdam: North-Holland; 1990.

[16] Duxbury PM, Beale PD, Leath PL. Size effects of electrical

breakdown in quenched random media. Phys Rev Lett 1986;57:

1052–5.

[17] Duxbury PM, Leath PL. Exactly solvable models of material

breakdown. Phys Rev B 1994;49:12676–87.

[18] Vanmarcke E. Random fields. Cambridge: MIT Press; 1983.

G. Frantziskonis / Probabilistic Engineering Mechanics 17 (2002) 349–357 357