- Email: [email protected]
A new fractal model for anisotropic surfaces
Pergamon Plh
A
NEW
FRACTAL
Int. J, Mach. Tools Manufact. Vol. 38. Nos 5-6. pp. 551-557, 1998 i 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0890-6955(97)00101-6 0890-6955 98 $19.00 + 0.00
MODEL
FOR
ANISOTROPIC
SURFACES
D. BLACKMOREt AND G. ZHOU~t
~Department of Mathematics, New Jersey Institute of Technology, Newark, New Jersey, USA $Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, USA
ABSTRACT: A new fractal-based functional model for anisotropic rough surfaces is used to devise and test two methods for the approximate computation of the fractal dimension of surfaces, and as an instrument for simulating the topography of engineering surfaces. A certain type of statistical self-affinity is proved for the model, and this property serves as the basis for one of the methods of approximating fractal dimension. The other technique for calculating fractal dimension is derived from a Htflder type condition satisfied by the model. Algorithms for implementing both of these new schemes for computing approximate values of fractal dimension are developed and compared with standard procedures. Both the functional model and its corresponding modified Gaussian height distribution are used for simulating fractal surfaces and several examples are adduced that strongly resemble some common anisotropic engineering surfaces. ~ 1998 Elsevier Scicncc Ltd
1 Introduction
Rough surface analysis and characterization play key roles in several areas of engineering and science including the design of superconductors, machine design, materials science, scattering of electromagnetic waves, surface contact and wear mechanics and tribology. The investigation of topography of engineering surfaces has a rather long and distinguished history; more recent developments in this field have been profoundly influenced by the statistically-basedformulations of LonguetHiggins[1] and Nayak[2], among others. But statisticalmethods have several shortcomings (such as nonstationarity of distributions,multiscales and instrument dependence of measurements), and this has lately given rise to an alternative approach to the analysis of engineering surfaces using fractal geometry [3,4]. This fractal approach, fueled by a number of notable successes over the last dozen years such as those reported by Mandelbrot[5], Gagnepain & Roques-Carmes[6], Thomas[7], Thomas & Thomas[8], Roques-Carmes et al.[9], Ling[10] and Majumdar & Tien[ll], has now become quite popular for investigations which have led to some useful advances in the theory and applications of rough surfaces as, for example, in Zhou et al.[12, 13] and Lopez et al.[14]. The Weierstrass-Mandelbrot function studied in Berry & Lewis[15], with its inherent fractal nature, has served as a starting point for many informative and useful investigations of rough surfaces via their profiles. These surface profiles represent the variation of surface topography along one-dimensional intersections of the surface with planes perpendic-
551
ular to the planar domains over which the surface is defined. In [16], we postulated a generalized Weierstrass-Mandelbrot type of surface profile model that possesses most of the known theoretical and physical features found in engineering surfaces, and using this model we were able to prove that the probability density function for the surface height distribution along a profile must be of the form of a Gaussian distribution multiplied by a convergent power series expanded about the mean height. If a surface is isotropic, a surface profile through a point suffices to completely characterize the local surface topography, but this is certainly not the case for anisotropic surfaces. Yet, the surface height distribution of anisotropic surfaces has also been observed to have the same type of slightly biased Gaussian form exhibited by their profiles. We have just developed [17] a fractal-based functional model that captures most of the features of anisotropic rough surfaces and from this proved that the height distribution is of the same type as that obtained for surface profiles [16]. In this paper we shall show how certain properties of our new model for anisotropic surfaces such as a form of self-attlnity and a HOlder type condition can be used for the approximate calculation of the fractal dimension, and we shall demonstrate how the model and its height distribution can be applied to the simulation of engineering and natural surfaces. We begin in Section 2 with a brief description of the functional model and its corresponding surface height distribution. Next, in Section 3, we give a proof of the exact sense in which the model is self-affine. In Section 4 we describe how the self-afflnity and H~lder properties of the surface function can be employed to compute the
552
D. Blackmore and G. Zhou
fractal dimension of anisotropic surfaces, and we compare these new methods with a standard technique for computing fractal dimension based upon the definition of the box dimension. Next, we show in Section 5 how our model can be used to simulate anisotropic surfaces and discuss the results of some examples of these simulations. We conclude in Section 6 with a few remarks on the significance of the results of this paper and some thoughts on future research. 2
The Model
Let R denote the real numbers, R 2 := {(x,y) : x,y E IR} be the plane and ]R3 := {(x,y,z) : x , y , z E R} denote Euclidean 3-space. The surface is assumed to have the form of a graph of a continuous function; i.e., S := graphO = {(x,y,O(x,y)): (x,y) C R},
(1)
]R2 and the integral is also Lebesgue. It can be proved that the box dimension of S, denoted by dimB(S), is equal to s when fl is sufficiently large [17], and the same is probably true of the Hausdorff dimension. Both of these dimensions will be assumed to be equal and referred to as the fractal dimension. The anisotropy of the surface (2) at a point X is generated by the anisotropy of the matrix A(X) whose strength or degree can be measured by the amount that A(X) (cos 0, sin 0)r varies as 0 ranges from 0 to 2n. Taking the probability measure on R to be P := (4ab) lm, we define the cumulative density function (CDF) to be (cf. Hoel [18])
F(z) := P ( { X : X e R, ~ ( X ) _< z}).
In [17] we proved that it follows from (S1)-(Sh) that the surface height distribution has a probability density function (PDF) satisfying
where R := [-a,a] × [-b,b] = { ( x , y ) : [xl <_ a, lYl < b, 0 < a, b} is a rectangular region and the function (I) : R --* R is continuous. More specifically, we postulate t h a t the surface height is z = +(x,y) = +(X) =
(5)
F(z) = /
f(t)dt,
--oo
of the form
OO
=~s-~y~B(~-a)"~(Y~A(X)x + r.),
(2)
f(z) = - ~ - ~ e
n=l
where a,13 and s are positive constants, X : = (x,Y) T, Fn := ("fn,/~n) T, where T denotes the transpose, qa is a continuous function and A is a matrixvalued function subject to the following properties: (S1) The parameters a , / 3 and s satisfy 0 < a , 1 < /7 and 2 _< s < 3, and the real sequences {%} and {un} are bounded. ($2) ~ : R2 + R is a continuous, piecewise smooth, doubly-periodic function. (S3) The surjective (onto) function ~ : IR2 [rain ~, max ~] has among its infinity of right inverses a continuous one ~* : [min ~, max ~] --* ~2. ($4) A ( X ) is a smooth, 2x2 matrix-valued function whose eigenvalues )~(X) satisfy r l _< IA(X)I < r2 for all X E ]R2 and some fixed 0 < r l < r2. Moreover, A satisfies the technical condition
maxR IIA'(X)XII -< em~n IIA(X)II ,
(a)
where IHI is the standard matrix norm and 0
¢(u)du
m ( X • n : ~ ( X ) < z}
~(R)
where m is the Lebesgue measure on
,
(4)
w, , ,,=o t . . v ~ )
(6)
where # -- E((b) is the mean and ~2 = E(((I) - p ) 2 ) is the variance of the height distribution, the power series as an infinite radius of convergence, and f, besides being smooth, is integrable and squareintegrable on tR. Here E denotes, as usual, the expected value operator. It can be shown that the statistical parameters p and a and all the coefficients wn depend in a fundamental (but rather complicated) way on the (fractal) parameters a, fl and s of the model (2). To conclude this summary of the properties of the anisotropic rough surface model, we note that in the process of computing the fractal dimension we were also able to prove t h a t the function (2) under assumptions (S1)-(Sh) also satisfies the HSlder condition
I ¢ ( x + h) - v ( x ) l [[h[13_, :=
O(X,h)
(7)
is a positive continuous function uniformly bounded away from zero on R for all sufficiently small ]lhll (cf. [14]). We note here that (7) also provides a measure of anisotropy of the surface in terms of the variation of O (X, [[hi[ (cos 0, sin 0) T) with 8 for small values of ][h[I. 3
Approximate
Self-Affinity
Although the model (2) is not exactly self-affine, we showed[17] using an essentially intuitive argument
A new fractal model that it possesses a type of approximate self-affnity that we denote by
~ ( ~ x ) ~_ fl3-'~(x). As we plan to use such a property as the basis for a procedure for calculating the fractal dimension of an isotropic rough surface, we shall formulate this attribute in more rigorous terms in this section. We shall require another assumption for our model in order to obtain a precise description of the kind of self-affinity that is appropriate for rough anisotropic surfaces. In particular, we also require that (2) satisfies the following property: b~
(S6) fl-2 /
a~
/
~o ( A ( f l - l X ) X
+ F 1 ) d X ---~0
-bl~-aft as ~----*oo. For example, we note that ~(x, y) = sin x cos y satisfies this assumption as well as ($2)-($5). For convenience, we define W := ¢ ( j 3 X ) fl3-s~(X). This function can be written as W = -a~-2qo 0 3 A ( X ) + F1) +
a.-2 ~'/3(s-3).[~o(ff~+lA(~X)X ~(fln+IA(X)X
+ r.)-
(8)
+ r.+,].
The uniform convergence of the series in (2) - that follows from (S1) and ($2) - implies that the average (or expected) value of W on R is
1// b
E ( W ) := ~
--
4ab
Thus, we see from our model assumptions (with ($6) playing a key role) that E ( W ) --* 0 as 13 --* oo. It seems reasonable to refer to this property as azyrnptotic statistical self-ajfinity ( A S S A ) and denote it by ¢ ( f l X ) ,~ fl3-sg2(X) as fl --~ oo. (11) One should compare this with the serf-affinity characterization in [14].
4
Computation of F r a c t a l D i m e n s i o n
In this section we shall show how the ASSA and HOlder properties of the model (2) can be used as bases for procedures for the approximate computation of the fractal dimension of an anisotropic rough surface. We shall also compare our new methods with a well-known algorithm for approximating fractal dimension that has its roots in the definition of the box dimension. There are now several standard methods for approximating the fractal dimension of a surface profile (see, for example [3]-[9] and [12]-[14]), but one of the simplest approaches follows straight from the definition of the box dimension. It works essentially as follows: Tessellate R 3 using a mesh consisting of identical cubes of very small side length e > 0. Then sample a large number of points on the surface S over a mesh on R of diameter considerably less than e, and let N(e) denote the total number of cubes of the e-tessellation containing sampled surface points. Whence
WdX =
(12)
is a reasonably good approximation of the box dimension that improves in accuracy with decreasing E.
We first describe how the HOlder property (7) may be used to approximate the fractal dimension of a surface. Define A := ](}(X + h) - ~ ( X ) ] , and note that upon taking the natural logarithm of (7) we obtain
~o(~A(X)X + rDdX+ b
oo
a
°'-'"-'V..<'-"<"-" fi n=l
log A = log O + (3 - s) log Ilhll.
-b -a
(9/
r . ) - ~o(~'+~A(X)X + r.+l)] dX. Making the change of variables Y = ~ X in the first integral on the right-hand side of (9), we obtmn
E ( W ) = - 4abe, - - 2
+ Fn+l)] d X .
logN(e) log(e -1 )
--b - G
~s-2
Fn) - ~(ff~+x A ( X ) X
~
--b-a
°.-.//b a
553
b~
a~
/
/ ~(A(~-XY)Y + rl)dY+
(13)
Setting U = log Ilhll and V = logA, we see that the graph of (13) in the U, V - p l a n e should approach a straight line with slope 3 - s as Hhll ---, 0. This suggests a procedure for estimating fractal dimension that we shall call the H - m e t h o d ; an approach that can be described as follows: First, select at random a base point Xo = (xo,yo) on the domain of the surface and measure
-b/~ - a f t
z0 := ~(X0). o¢)
b
ocS--2fl3--s2fl(s-3)(n--1) f f n=l -b --a
The remaining data is sampled on a polar mesh centered at X0. For this, we define
(101
X0 + h := X0 + r(cos 0, sin 0),
554
D. Blackmore and G. Zhou
and select rays 0 = 0i := 27ri/rn, i = O , . . . , m - 1, and radial distances r j = j r . ~ n , j = 1, ...,n, for some small r . > 0. In-order to collect a sufficient amount of data, m and n should be reasonably large positive integers. We next measure
O(X) _> 1 for all X E R, since this can be achieved by a simple translation of the coordinate system along the z - a x i s . Next, we compute the averages m--1
1 Ek := -m-n E
zij = • (Xo + rj (cos 0i, sin Oi))
for0
1
i=0
andset
A~j := Iz~j - z 0 1 . W i t h this data, (13) provides a natural instrument for approximating the fractal dimension s beginning with the computation of the means rn--]
n
EEijk
(17)
j=l
for 2 < k < q. According to the ASSA property, approximately embodied in (15) or (16), we expect the d a t a points {(~k,'--k)} to nearly lie on a line when plotted on log-log paper. Hence, we determine the least-squares line corresponding to the d a t a {(log~k,logEk ) : 2 < k < q} and compute its slope LA. We can then estimate the fractal dimension of a rough anisotropic surface by s ~- 3 - LA.
(18)
i=0
for 1 < j < n. After this, we plot the d a t a points { ( r j , z ~ ) } o n log-log paper and select a positive integer n . , with say 5 < n. < n, such that the first n . points of the d a t a set exhibit a strong linear tendency. Then we determine the least-squares line fitting the d a t a { ( l o g r j , l o g A j ) : 1 < j <_ n . } and compute the slope LH of this line. It follows from (13) that the fractal dimension can be approximated by s ~-- 3 - L H . (14) Another technique for approximating the fractal dimension of an engineering surface is one t h a t exploits the ASSA property; and we shall call this the A-method. Again we randomly select a point X0 on the domain of the surface and introduce a polar mesh of points
Xij :-~ Xo ~- rj(cosOi,Sin Oi), where 0i = 27ri/rn, 0 < i < m - 1, and rj = j r # / n , 1 < j < n, for some r # > 0. But this time we make sure t h a t r # is so small that the disk of radius qr# centered at X0 is also contained in the domain R of the surface for some fairly large positive integer q. Since we are only estimating here, we may recast the ASSA property in the form of the equality (for fl large) z = ~3-~, (15) where ~. = .=.(j3) := average ( ~ ( I ~ X ) / O ( X ) ) taken over an appropriate region. Upon taking the natural logarithm of (15), we obtain
Observe t h a t in contradistinction to the box dimension algorithm, both the H-method and the Amethod are inherently local. It is possible, however, that these new methods might be improved by some degree of global averaging. 5
Simulation of Surfaces
Our fractal functional model and its corresponding height distribution also provide convenient means for creating fractal surfaces. We shall illustrate two ways in which this can be accomplished in this section. The functional model (2) can be employed directly to simulate anisotropic surfaces by simply making a choice of the parameters, the function ~o and the matrix A consistent with the assumptions. We illustrate this with an example. 5.1 Example Choose ~o(x,y) = c o s x c o s y , a = 1, j3 = 4, s = 2.5, Fn = (0, 1) T for all positive integers n, R = { X :O < x, y g l } and A(X) =
(1
0
0.1 sin(xy) ) -2
so that the functional model has the form oo
• (x,y)
=
~2-"eos{4~[x+0.1ysin(~)]}. cos[1 - 2.4"y].
logE = (3 - s)logiC.
we can now use (16) for our approximate calculation. Consider values of ~ = ilk, where 2 < k < q and measure
--~k -
(19)
(16)
¢(~kx~) ¢(x~j)
for 0 < i _ m - l , 1 < j < n, 2 _< k < q. at this point we note t h a t it may be assumed t h a t
The graph of (19) in Figure 1 reveals both the fractal and anisotropic natures of the surface. In fact, the surface in this example is typical of grinding surfaces that tend to exhibit fractal lineaments coupled with a strong directional pattern (note the ridges along the y direction). In our next example we shall show how the height distribution (6) can be used to obtain a MonteCarlo simulation of an engineering surface. Note that once the mean tt and the variance a 2 in (6)
A new fractal model
555
2
N O.
O, -0.5-1;
2
2 •
,
o..................~% y
FIG. 1.
4
0
'.
0
•
x
Simulation of anisotropic fractal surface of Example 5.1.
~
3.
I
2N
1 0 -1
I
!
I
-2 2
2
1
0
0
-1
.
-2
FIG. 2.
-2
x
Monte-Carlo simulation of fractal surface in Example 5.2.
556
D. Blackmore and G. Zhou
are specified, the coefficients {wn} cannot be chosen arbitrarily since f must satisfy consistency constraints dictated by the moments of the distribution, such as
f
f ( z ) d z = 1, f z f ( z ) d z = # ,
--00
--~
(20)
/ (z - ,)2 f ( z ) d z = ~2 ...
5.2 Example Select tt = 0 a n d a = 1 in (6). Then it can be readily verified that the PDF
f(z)=~e
1
-z2/2(
-
In
z+2z 2-5z
-
1
) z4
(21)
satisfies the necessary consistency requirements (20). Integrating (21), we obtain the corresponding CDF
f(z)
=
l~:-.-[(Z3 + Z2 -- 3Z -- 1) + 3 X/2~
3 ~f e-t~/2dt].
(22)
We form the 61×61 mesh of points on R = {(x,y) : - 2 <_ x,y <_ 2} using the subdivision points (~o,~), 0 <_ i,j <_ 60. At each of these points we choose a random number in the interval 0 < w < 1 using a pseudo-random number generator. Then if the random number selected at the point ( ~ , ~ ) is wij, it follows that the corresponding value of the surface height is
zij = F-1 (Wij), where F -1 is the inverse of the function defined by (22). The results of one such Monte-Carlo simulation are graphed in Figure 2. It can be seen from this figure that the simulated topography not only exhibits the unmistakable features of a general random surface, but is also imbued with intrinsic anisotropic and fxactal attributes.
6
Concluding Remarks
We have demonstrated how our fractal model and its surface height distribution can be used to estimate the fractal dimension of a rough anisotropic surface and even simulate such a surface. The procedures developed for approximating fractal dimensions appear to compare rather well with more familiar methods, but it will require further investigation, including comparisons of actual calculations for several anisotropic engineering surfaces, to
obtain a more definitive grasp of the relative ad. vantages and disadvantages of these new schemes Questions regarding the accuracy and robustnes, (which might be enhanced by averaging over severa: base points rather than a single randomly selected point) of the methods introduced and the computational cost of their attendant calculations have bee~ essentially ignored here and await a more thoroug~ investigation. The direct and Monte-Carlo simulation techniques that we have described seem to wor~ quite well, and it may be useful to inquire into the possibility of integrating properties of actual surfaces directly into the functional representations foI the purpose of improving the applicability of these methods to engineering practice. REFERENCES [1] M. Longuet-Higgins, The Statistical Analysis of a Random Moving Surface, Phil. Trans. R. Soc. London, A248, 321-387 (1957). [2] P. Nayak, Some Aspects of Surface Roughness Measurement, Wear, 26, 165-174 (1973). [3] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York (1990). [4] K. Falconer, Techniques in Fractal Geometry, Wiley, New York (1996). [5] B. Mandelbrot, Self-A~ine Fractals and Fractal Dimension, Physica Scripta, 32, 257-260 (1985). [6] J. Gagnepain, and C. Roques-Carmes, Frac-
tal Approach to Two-Dimensional and ThreeDimensional Surface Roughness, Wear, 109, 119126 (1986). [7] T. Thomas, Surface Roughness: The Next Ten Years, Surface Topography, 1, 3-9 (1988). [8] A. Thomas, and T. Thomas, Digital Analysis of Very Small Scale Surface Roughness, J. WaveMaterial Interaction, 341-350 (1988). [9] C. Rouques-Carmes, D. Wehbi, J. Quiniou, and C. Tricot, Modeling Engineering Surfaces and
Evaluating Their Non-Integer Dimension in Material Science, Surface Topography, 1,435-443
(1988). [10] F. Ling, Fraetals, Engineering Surfaces and Tribology, Wear, 136, 141-156 (1990). [11] A. Majumdar, and C. Tien, Fraetal Character-
ization and Simulation of Rough Surfaces, Wear, 136, 313-327 (1990). [12] G. Zhou, M. C. Leu, and D. Blackmore, Fractal Geometry Model for Wear Prediction, Wear, 170/1, 1-14 (1993). [13] G. Zhou, M. C. Leu, and D. Blackmore, Fractal
Geometry Modeling with Applications in Surface Characterization and Wear Prediction, J. Mach. Tools Manufact., 35, 203-209 (1995). [14] J. Lopez, G. Hansali, J. Le Bosse, and T. Mathia, Caracterisation Fractale de la Rugosire Tridimensionelle d'une Surface, J. de Physique III, 4, 2501-2519 (1994). [15] M. Berry, and Z. Lewis, On the WeierstrassMandelbrot Fractal Function, Proc. Royal Soc., A 370, 459-484 (1980).
A new fractal model [16] D. Blackmore, and G. Zhou, A General Fractal Distribution Function for Rough Surface Proles, SIAM J. Appl. Math., 56, 1694-1719 1996a).
~
557
[17] D. Blackmore, and G. Zhou, Fractal Analysis of Height Distributions of Anisotropic Rough Surfaces, Wear (submitted: 1996b). [18] P. Hoel, Introduction to Mathematical Statistics, Wiley, New York (1971).
A
NEW
FRACTAL
Int. J, Mach. Tools Manufact. Vol. 38. Nos 5-6. pp. 551-557, 1998 i 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0890-6955(97)00101-6 0890-6955 98 $19.00 + 0.00
MODEL
FOR
ANISOTROPIC
SURFACES
D. BLACKMOREt AND G. ZHOU~t
~Department of Mathematics, New Jersey Institute of Technology, Newark, New Jersey, USA $Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, USA
ABSTRACT: A new fractal-based functional model for anisotropic rough surfaces is used to devise and test two methods for the approximate computation of the fractal dimension of surfaces, and as an instrument for simulating the topography of engineering surfaces. A certain type of statistical self-affinity is proved for the model, and this property serves as the basis for one of the methods of approximating fractal dimension. The other technique for calculating fractal dimension is derived from a Htflder type condition satisfied by the model. Algorithms for implementing both of these new schemes for computing approximate values of fractal dimension are developed and compared with standard procedures. Both the functional model and its corresponding modified Gaussian height distribution are used for simulating fractal surfaces and several examples are adduced that strongly resemble some common anisotropic engineering surfaces. ~ 1998 Elsevier Scicncc Ltd
1 Introduction
Rough surface analysis and characterization play key roles in several areas of engineering and science including the design of superconductors, machine design, materials science, scattering of electromagnetic waves, surface contact and wear mechanics and tribology. The investigation of topography of engineering surfaces has a rather long and distinguished history; more recent developments in this field have been profoundly influenced by the statistically-basedformulations of LonguetHiggins[1] and Nayak[2], among others. But statisticalmethods have several shortcomings (such as nonstationarity of distributions,multiscales and instrument dependence of measurements), and this has lately given rise to an alternative approach to the analysis of engineering surfaces using fractal geometry [3,4]. This fractal approach, fueled by a number of notable successes over the last dozen years such as those reported by Mandelbrot[5], Gagnepain & Roques-Carmes[6], Thomas[7], Thomas & Thomas[8], Roques-Carmes et al.[9], Ling[10] and Majumdar & Tien[ll], has now become quite popular for investigations which have led to some useful advances in the theory and applications of rough surfaces as, for example, in Zhou et al.[12, 13] and Lopez et al.[14]. The Weierstrass-Mandelbrot function studied in Berry & Lewis[15], with its inherent fractal nature, has served as a starting point for many informative and useful investigations of rough surfaces via their profiles. These surface profiles represent the variation of surface topography along one-dimensional intersections of the surface with planes perpendic-
551
ular to the planar domains over which the surface is defined. In [16], we postulated a generalized Weierstrass-Mandelbrot type of surface profile model that possesses most of the known theoretical and physical features found in engineering surfaces, and using this model we were able to prove that the probability density function for the surface height distribution along a profile must be of the form of a Gaussian distribution multiplied by a convergent power series expanded about the mean height. If a surface is isotropic, a surface profile through a point suffices to completely characterize the local surface topography, but this is certainly not the case for anisotropic surfaces. Yet, the surface height distribution of anisotropic surfaces has also been observed to have the same type of slightly biased Gaussian form exhibited by their profiles. We have just developed [17] a fractal-based functional model that captures most of the features of anisotropic rough surfaces and from this proved that the height distribution is of the same type as that obtained for surface profiles [16]. In this paper we shall show how certain properties of our new model for anisotropic surfaces such as a form of self-attlnity and a HOlder type condition can be used for the approximate calculation of the fractal dimension, and we shall demonstrate how the model and its height distribution can be applied to the simulation of engineering and natural surfaces. We begin in Section 2 with a brief description of the functional model and its corresponding surface height distribution. Next, in Section 3, we give a proof of the exact sense in which the model is self-affine. In Section 4 we describe how the self-afflnity and H~lder properties of the surface function can be employed to compute the
552
D. Blackmore and G. Zhou
fractal dimension of anisotropic surfaces, and we compare these new methods with a standard technique for computing fractal dimension based upon the definition of the box dimension. Next, we show in Section 5 how our model can be used to simulate anisotropic surfaces and discuss the results of some examples of these simulations. We conclude in Section 6 with a few remarks on the significance of the results of this paper and some thoughts on future research. 2
The Model
Let R denote the real numbers, R 2 := {(x,y) : x,y E IR} be the plane and ]R3 := {(x,y,z) : x , y , z E R} denote Euclidean 3-space. The surface is assumed to have the form of a graph of a continuous function; i.e., S := graphO = {(x,y,O(x,y)): (x,y) C R},
(1)
]R2 and the integral is also Lebesgue. It can be proved that the box dimension of S, denoted by dimB(S), is equal to s when fl is sufficiently large [17], and the same is probably true of the Hausdorff dimension. Both of these dimensions will be assumed to be equal and referred to as the fractal dimension. The anisotropy of the surface (2) at a point X is generated by the anisotropy of the matrix A(X) whose strength or degree can be measured by the amount that A(X) (cos 0, sin 0)r varies as 0 ranges from 0 to 2n. Taking the probability measure on R to be P := (4ab) lm, we define the cumulative density function (CDF) to be (cf. Hoel [18])
F(z) := P ( { X : X e R, ~ ( X ) _< z}).
In [17] we proved that it follows from (S1)-(Sh) that the surface height distribution has a probability density function (PDF) satisfying
where R := [-a,a] × [-b,b] = { ( x , y ) : [xl <_ a, lYl < b, 0 < a, b} is a rectangular region and the function (I) : R --* R is continuous. More specifically, we postulate t h a t the surface height is z = +(x,y) = +(X) =
(5)
F(z) = /
f(t)dt,
--oo
of the form
OO
=~s-~y~B(~-a)"~(Y~A(X)x + r.),
(2)
f(z) = - ~ - ~ e
n=l
where a,13 and s are positive constants, X : = (x,Y) T, Fn := ("fn,/~n) T, where T denotes the transpose, qa is a continuous function and A is a matrixvalued function subject to the following properties: (S1) The parameters a , / 3 and s satisfy 0 < a , 1 < /7 and 2 _< s < 3, and the real sequences {%} and {un} are bounded. ($2) ~ : R2 + R is a continuous, piecewise smooth, doubly-periodic function. (S3) The surjective (onto) function ~ : IR2 [rain ~, max ~] has among its infinity of right inverses a continuous one ~* : [min ~, max ~] --* ~2. ($4) A ( X ) is a smooth, 2x2 matrix-valued function whose eigenvalues )~(X) satisfy r l _< IA(X)I < r2 for all X E ]R2 and some fixed 0 < r l < r2. Moreover, A satisfies the technical condition
maxR IIA'(X)XII -< em~n IIA(X)II ,
(a)
where IHI is the standard matrix norm and 0
¢(u)du
m ( X • n : ~ ( X ) < z}
~(R)
where m is the Lebesgue measure on
,
(4)
w, , ,,=o t . . v ~ )
(6)
where # -- E((b) is the mean and ~2 = E(((I) - p ) 2 ) is the variance of the height distribution, the power series as an infinite radius of convergence, and f, besides being smooth, is integrable and squareintegrable on tR. Here E denotes, as usual, the expected value operator. It can be shown that the statistical parameters p and a and all the coefficients wn depend in a fundamental (but rather complicated) way on the (fractal) parameters a, fl and s of the model (2). To conclude this summary of the properties of the anisotropic rough surface model, we note that in the process of computing the fractal dimension we were also able to prove t h a t the function (2) under assumptions (S1)-(Sh) also satisfies the HSlder condition
I ¢ ( x + h) - v ( x ) l [[h[13_, :=
O(X,h)
(7)
is a positive continuous function uniformly bounded away from zero on R for all sufficiently small ]lhll (cf. [14]). We note here that (7) also provides a measure of anisotropy of the surface in terms of the variation of O (X, [[hi[ (cos 0, sin 0) T) with 8 for small values of ][h[I. 3
Approximate
Self-Affinity
Although the model (2) is not exactly self-affine, we showed[17] using an essentially intuitive argument
A new fractal model that it possesses a type of approximate self-affnity that we denote by
~ ( ~ x ) ~_ fl3-'~(x). As we plan to use such a property as the basis for a procedure for calculating the fractal dimension of an isotropic rough surface, we shall formulate this attribute in more rigorous terms in this section. We shall require another assumption for our model in order to obtain a precise description of the kind of self-affinity that is appropriate for rough anisotropic surfaces. In particular, we also require that (2) satisfies the following property: b~
(S6) fl-2 /
a~
/
~o ( A ( f l - l X ) X
+ F 1 ) d X ---~0
-bl~-aft as ~----*oo. For example, we note that ~(x, y) = sin x cos y satisfies this assumption as well as ($2)-($5). For convenience, we define W := ¢ ( j 3 X ) fl3-s~(X). This function can be written as W = -a~-2qo 0 3 A ( X ) + F1) +
a.-2 ~'/3(s-3).[~o(ff~+lA(~X)X ~(fln+IA(X)X
+ r.)-
(8)
+ r.+,].
The uniform convergence of the series in (2) - that follows from (S1) and ($2) - implies that the average (or expected) value of W on R is
1// b
E ( W ) := ~
--
4ab
Thus, we see from our model assumptions (with ($6) playing a key role) that E ( W ) --* 0 as 13 --* oo. It seems reasonable to refer to this property as azyrnptotic statistical self-ajfinity ( A S S A ) and denote it by ¢ ( f l X ) ,~ fl3-sg2(X) as fl --~ oo. (11) One should compare this with the serf-affinity characterization in [14].
4
Computation of F r a c t a l D i m e n s i o n
In this section we shall show how the ASSA and HOlder properties of the model (2) can be used as bases for procedures for the approximate computation of the fractal dimension of an anisotropic rough surface. We shall also compare our new methods with a well-known algorithm for approximating fractal dimension that has its roots in the definition of the box dimension. There are now several standard methods for approximating the fractal dimension of a surface profile (see, for example [3]-[9] and [12]-[14]), but one of the simplest approaches follows straight from the definition of the box dimension. It works essentially as follows: Tessellate R 3 using a mesh consisting of identical cubes of very small side length e > 0. Then sample a large number of points on the surface S over a mesh on R of diameter considerably less than e, and let N(e) denote the total number of cubes of the e-tessellation containing sampled surface points. Whence
WdX =
(12)
is a reasonably good approximation of the box dimension that improves in accuracy with decreasing E.
We first describe how the HOlder property (7) may be used to approximate the fractal dimension of a surface. Define A := ](}(X + h) - ~ ( X ) ] , and note that upon taking the natural logarithm of (7) we obtain
~o(~A(X)X + rDdX+ b
oo
a
°'-'"-'V..<'-"<"-" fi n=l
log A = log O + (3 - s) log Ilhll.
-b -a
(9/
r . ) - ~o(~'+~A(X)X + r.+l)] dX. Making the change of variables Y = ~ X in the first integral on the right-hand side of (9), we obtmn
E ( W ) = - 4abe, - - 2
+ Fn+l)] d X .
logN(e) log(e -1 )
--b - G
~s-2
Fn) - ~(ff~+x A ( X ) X
~
--b-a
°.-.//b a
553
b~
a~
/
/ ~(A(~-XY)Y + rl)dY+
(13)
Setting U = log Ilhll and V = logA, we see that the graph of (13) in the U, V - p l a n e should approach a straight line with slope 3 - s as Hhll ---, 0. This suggests a procedure for estimating fractal dimension that we shall call the H - m e t h o d ; an approach that can be described as follows: First, select at random a base point Xo = (xo,yo) on the domain of the surface and measure
-b/~ - a f t
z0 := ~(X0). o¢)
b
ocS--2fl3--s2fl(s-3)(n--1) f f n=l -b --a
The remaining data is sampled on a polar mesh centered at X0. For this, we define
(101
X0 + h := X0 + r(cos 0, sin 0),
554
D. Blackmore and G. Zhou
and select rays 0 = 0i := 27ri/rn, i = O , . . . , m - 1, and radial distances r j = j r . ~ n , j = 1, ...,n, for some small r . > 0. In-order to collect a sufficient amount of data, m and n should be reasonably large positive integers. We next measure
O(X) _> 1 for all X E R, since this can be achieved by a simple translation of the coordinate system along the z - a x i s . Next, we compute the averages m--1
1 Ek := -m-n E
zij = • (Xo + rj (cos 0i, sin Oi))
for0
1
i=0
andset
A~j := Iz~j - z 0 1 . W i t h this data, (13) provides a natural instrument for approximating the fractal dimension s beginning with the computation of the means rn--]
n
EEijk
(17)
j=l
for 2 < k < q. According to the ASSA property, approximately embodied in (15) or (16), we expect the d a t a points {(~k,'--k)} to nearly lie on a line when plotted on log-log paper. Hence, we determine the least-squares line corresponding to the d a t a {(log~k,logEk ) : 2 < k < q} and compute its slope LA. We can then estimate the fractal dimension of a rough anisotropic surface by s ~- 3 - LA.
(18)
i=0
for 1 < j < n. After this, we plot the d a t a points { ( r j , z ~ ) } o n log-log paper and select a positive integer n . , with say 5 < n. < n, such that the first n . points of the d a t a set exhibit a strong linear tendency. Then we determine the least-squares line fitting the d a t a { ( l o g r j , l o g A j ) : 1 < j <_ n . } and compute the slope LH of this line. It follows from (13) that the fractal dimension can be approximated by s ~-- 3 - L H . (14) Another technique for approximating the fractal dimension of an engineering surface is one t h a t exploits the ASSA property; and we shall call this the A-method. Again we randomly select a point X0 on the domain of the surface and introduce a polar mesh of points
Xij :-~ Xo ~- rj(cosOi,Sin Oi), where 0i = 27ri/rn, 0 < i < m - 1, and rj = j r # / n , 1 < j < n, for some r # > 0. But this time we make sure t h a t r # is so small that the disk of radius qr# centered at X0 is also contained in the domain R of the surface for some fairly large positive integer q. Since we are only estimating here, we may recast the ASSA property in the form of the equality (for fl large) z = ~3-~, (15) where ~. = .=.(j3) := average ( ~ ( I ~ X ) / O ( X ) ) taken over an appropriate region. Upon taking the natural logarithm of (15), we obtain
Observe t h a t in contradistinction to the box dimension algorithm, both the H-method and the Amethod are inherently local. It is possible, however, that these new methods might be improved by some degree of global averaging. 5
Simulation of Surfaces
Our fractal functional model and its corresponding height distribution also provide convenient means for creating fractal surfaces. We shall illustrate two ways in which this can be accomplished in this section. The functional model (2) can be employed directly to simulate anisotropic surfaces by simply making a choice of the parameters, the function ~o and the matrix A consistent with the assumptions. We illustrate this with an example. 5.1 Example Choose ~o(x,y) = c o s x c o s y , a = 1, j3 = 4, s = 2.5, Fn = (0, 1) T for all positive integers n, R = { X :O < x, y g l } and A(X) =
(1
0
0.1 sin(xy) ) -2
so that the functional model has the form oo
• (x,y)
=
~2-"eos{4~[x+0.1ysin(~)]}. cos[1 - 2.4"y].
logE = (3 - s)logiC.
we can now use (16) for our approximate calculation. Consider values of ~ = ilk, where 2 < k < q and measure
--~k -
(19)
(16)
¢(~kx~) ¢(x~j)
for 0 < i _ m - l , 1 < j < n, 2 _< k < q. at this point we note t h a t it may be assumed t h a t
The graph of (19) in Figure 1 reveals both the fractal and anisotropic natures of the surface. In fact, the surface in this example is typical of grinding surfaces that tend to exhibit fractal lineaments coupled with a strong directional pattern (note the ridges along the y direction). In our next example we shall show how the height distribution (6) can be used to obtain a MonteCarlo simulation of an engineering surface. Note that once the mean tt and the variance a 2 in (6)
A new fractal model
555
2
N O.
O, -0.5-1;
2
2 •
,
o..................~% y
FIG. 1.
4
0
'.
0
•
x
Simulation of anisotropic fractal surface of Example 5.1.
~
3.
I
2N
1 0 -1
I
!
I
-2 2
2
1
0
0
-1
.
-2
FIG. 2.
-2
x
Monte-Carlo simulation of fractal surface in Example 5.2.
556
D. Blackmore and G. Zhou
are specified, the coefficients {wn} cannot be chosen arbitrarily since f must satisfy consistency constraints dictated by the moments of the distribution, such as
f
f ( z ) d z = 1, f z f ( z ) d z = # ,
--00
--~
(20)
/ (z - ,)2 f ( z ) d z = ~2 ...
5.2 Example Select tt = 0 a n d a = 1 in (6). Then it can be readily verified that the PDF
f(z)=~e
1
-z2/2(
-
In
z+2z 2-5z
-
1
) z4
(21)
satisfies the necessary consistency requirements (20). Integrating (21), we obtain the corresponding CDF
f(z)
=
l~:-.-[(Z3 + Z2 -- 3Z -- 1) + 3 X/2~
3 ~f e-t~/2dt].
(22)
We form the 61×61 mesh of points on R = {(x,y) : - 2 <_ x,y <_ 2} using the subdivision points (~o,~), 0 <_ i,j <_ 60. At each of these points we choose a random number in the interval 0 < w < 1 using a pseudo-random number generator. Then if the random number selected at the point ( ~ , ~ ) is wij, it follows that the corresponding value of the surface height is
zij = F-1 (Wij), where F -1 is the inverse of the function defined by (22). The results of one such Monte-Carlo simulation are graphed in Figure 2. It can be seen from this figure that the simulated topography not only exhibits the unmistakable features of a general random surface, but is also imbued with intrinsic anisotropic and fxactal attributes.
6
Concluding Remarks
We have demonstrated how our fractal model and its surface height distribution can be used to estimate the fractal dimension of a rough anisotropic surface and even simulate such a surface. The procedures developed for approximating fractal dimensions appear to compare rather well with more familiar methods, but it will require further investigation, including comparisons of actual calculations for several anisotropic engineering surfaces, to
obtain a more definitive grasp of the relative ad. vantages and disadvantages of these new schemes Questions regarding the accuracy and robustnes, (which might be enhanced by averaging over severa: base points rather than a single randomly selected point) of the methods introduced and the computational cost of their attendant calculations have bee~ essentially ignored here and await a more thoroug~ investigation. The direct and Monte-Carlo simulation techniques that we have described seem to wor~ quite well, and it may be useful to inquire into the possibility of integrating properties of actual surfaces directly into the functional representations foI the purpose of improving the applicability of these methods to engineering practice. REFERENCES [1] M. Longuet-Higgins, The Statistical Analysis of a Random Moving Surface, Phil. Trans. R. Soc. London, A248, 321-387 (1957). [2] P. Nayak, Some Aspects of Surface Roughness Measurement, Wear, 26, 165-174 (1973). [3] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York (1990). [4] K. Falconer, Techniques in Fractal Geometry, Wiley, New York (1996). [5] B. Mandelbrot, Self-A~ine Fractals and Fractal Dimension, Physica Scripta, 32, 257-260 (1985). [6] J. Gagnepain, and C. Roques-Carmes, Frac-
tal Approach to Two-Dimensional and ThreeDimensional Surface Roughness, Wear, 109, 119126 (1986). [7] T. Thomas, Surface Roughness: The Next Ten Years, Surface Topography, 1, 3-9 (1988). [8] A. Thomas, and T. Thomas, Digital Analysis of Very Small Scale Surface Roughness, J. WaveMaterial Interaction, 341-350 (1988). [9] C. Rouques-Carmes, D. Wehbi, J. Quiniou, and C. Tricot, Modeling Engineering Surfaces and
Evaluating Their Non-Integer Dimension in Material Science, Surface Topography, 1,435-443
(1988). [10] F. Ling, Fraetals, Engineering Surfaces and Tribology, Wear, 136, 141-156 (1990). [11] A. Majumdar, and C. Tien, Fraetal Character-
ization and Simulation of Rough Surfaces, Wear, 136, 313-327 (1990). [12] G. Zhou, M. C. Leu, and D. Blackmore, Fractal Geometry Model for Wear Prediction, Wear, 170/1, 1-14 (1993). [13] G. Zhou, M. C. Leu, and D. Blackmore, Fractal
Geometry Modeling with Applications in Surface Characterization and Wear Prediction, J. Mach. Tools Manufact., 35, 203-209 (1995). [14] J. Lopez, G. Hansali, J. Le Bosse, and T. Mathia, Caracterisation Fractale de la Rugosire Tridimensionelle d'une Surface, J. de Physique III, 4, 2501-2519 (1994). [15] M. Berry, and Z. Lewis, On the WeierstrassMandelbrot Fractal Function, Proc. Royal Soc., A 370, 459-484 (1980).
A new fractal model [16] D. Blackmore, and G. Zhou, A General Fractal Distribution Function for Rough Surface Proles, SIAM J. Appl. Math., 56, 1694-1719 1996a).
~
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[17] D. Blackmore, and G. Zhou, Fractal Analysis of Height Distributions of Anisotropic Rough Surfaces, Wear (submitted: 1996b). [18] P. Hoel, Introduction to Mathematical Statistics, Wiley, New York (1971).