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On effective transversely isotropic elasticity tensors based on Frobenius and L2 operator norms

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A generally anisotropic elasticity tensor, which might be obtained from physical measurements, can be approximated by a tensor belonging to a particular material-symmetry class; we refer to such a tensor as the effective tensor. The effective tensor is the closest to the generally anisotropic tensor among the tensors of that symmetry class. The concept of closeness is formalized in the notion of norm. Herein, we compare the effective tensors belonging to the transversely isotropic class and obtained using two different norms: the Frobenius norm and the L2 operator norm. We compare distributions of the effective elasticity parameters and symmetry-axis orientations for both the error-free case and the case of the generally anisotropic tensor subject to errors. © 2009 - 2014. Universita degli Studi di Padova - Padova University Press - All Rights Reserved.
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Volume 7 ·2014 ·Pages 1–6
On effective transversely isotropic elasticity tensors based on
Frobenius and L2operator norms
Tomasz Daneka·Michael A. Slawinski b
Abstract
A generally anisotropic elasticity tensor, which might be obtained from physical measurements, can be
approximated by a tensor belonging to a particular material-symmetry class; we refer to such a tensor
as the effective tensor. The effective tensor is the closest to the generally anisotropic tensor among the
tensors of that symmetry class. The concept of closeness is formalized in the notion of norm. Herein,
we compare the effective tensors belonging to the transversely isotropic class and obtained using two
different norms: the Frobenius norm and the
L2
operator norm. We compare distributions of the effective
elasticity parameters and symmetry-axis orientations for both the error-free case and the case of the
generally anisotropic tensor subject to errors.
1 Introduction
A Hookean solid is defined by the elasticity tensor,
ci jk`
, which stems from the constitutive relation that relates linearly the stress,
σi j , and strain, "k`, tensors,
σi j =
3
X
k,`=1
ci jk`"k`,i,j∈ {1, 2, 3}. (1)
The elasticity tensor is assumed to satisfy the index symmetries,
ci jk`
=
cjik`
=
ck`i j
, and be positive-definite; these requirements
reduce the number of independent components to twenty-one and, hence, allow us to represent equation (1) as
σ11
σ22
σ33
p2σ23
p2σ13
p2σ12
=
c1111 c1122 c1133 p2c1123 p2c1113 p2c1112
c1122 c2222 c2233 p2c2223 p2c2213 p2c2212
c1133 c2233 c3333 p2c3323 p2c3313 p2c3312
p2c1123 p2c2223 p2c3323 2c2323 2c2313 2c2312
p2c1113 p2c2213 p2c3313 2c2313 2c1313 2c1312
p2c1112 p2c2212 p2c3312 2c2312 2c1312 2c1212
"11
"22
"33
p2"23
p2"13
p2"12
. (2)
As shown by several researchers—among them Forte and Vianello [
6
]and Bóna et al. [
1
]—the elasticity tensor belongs to one of
eight material-symmetry classes, from general anisotropy to isotropy. It is common to approximate a generally anisotropic tensor
by its closest counterpart that belongs to particular symmetry classes. Such a tensor is referred to as the effective tensor of that
class. There might be several motivations for such an approximation; in view of uncertainties of
ci jk`
, we might require a model
with fewer parameters; also, a particular symmetry might allow us to recognize properties of the material represented by the
elasticity tensor. However, depending on the applied norm, we obtain different effective tensors of a given class.
In this paper, we consider the Frobenius norm and the operator norm induced by the
L2
norm on the spaces of the elasticity
tensor in expression (1) to examine the transversely isotropic effective tensors, whose generic form is
cTI =
cTI
1111 cTI
1122 cTI
1133 0 0 0
cTI
1122 cTI
1111 cTI
1133 0 0 0
cTI
1133 cTI
1133 cTI
3333 0 0 0
0 0 0 2cTI
2323 0 0
0 0 0 0 2cTI
2323 0
0 0 0 0 0 cTI
1111 cTI
1122
. (3)
In particular, we examine the effect of errors in the components of the generally anisotropic tensor on the parameters and
orientations of its transversely isotropic effective tensors.
2 Norms and distances for effective tensors
The Frobenius norm treats the square matrix in expression (2) as a Euclidean vector; it is the square root of the sum of squared
components,
ci jk`
. This norm exhibits the rotational invariance, which—in general—is convenient to study anisotropy; it is used
commonly in such studies (e.g., Kochetov and Slawinski [8], Danek et al. [4]).
aDepartment of Geoinformatics and Applied Computer Science, AGH - University of Science and Technology, Kraków, Poland
bDepartment of Earth Sciences, Memorial University of Newfoundland, St. John’s, NL, Canada
Danek ·Slawinski 2
As discussed by Bos and Slawinski [
2
], another matrix norm that exhibits the rotational invariance is the induced Euclidean
operator norm,
kAk2:=max
kxk2=1kAxk2,
which—for A=At, as is the case of the square matrix, C, in expression (2)—becomes
kAk2:=max{|λ|:λan eigenvalue of A}; (4)
we refer to this norm as the operator norm.
For the elasticity tensor, c, we let λ1,. . . , λ6be the eigenvalues of the corresponding matrix C; then, we write
kckF:=v
u
u
t
3
X
i,j,k,`=1
c2
i jk`=pλ2
1+··· +λ2
6, (5)
and
kck2:=max{|λi|:i=1, . . . , 6}, (6)
which are the Frobenius and operator norms, respectively.
The former corresponds to the quadratic average of the norm of the stress over strains whose norm is unity:
p6 avgk"k=1kC"k2
=
kckF. The latter is the maximum of that quantity: maxk"k=1kC"k=kck2.
In view of expression (1), one could consider an operator norm induced by various norms for the stress and strain tensors,
which are second-rank tensors; for example, one could consider the matrix operator
L2
norm or indeed another
Lp
norm.
However, since the resulting norm is not the operator
L2
norm of the matrix representation,
C
, of tensor
c
(L. Bos, pers.comm.,
2013), we do not consider it in this paper. Herein, the norm of the stress and strain tensors is defined as the Euclidean length of
their vectorial representations, as in expression (2).
The Frobenius-norm effective tensor,
ˆ
c
, relative to the fixed orientation is the orthogonal projection of the generally
anisotropic tensor,
c
, in the sense of the Frobenius norm, on the linear space containing all tensors of a given symmetry. This
projection is the average given by
ˆ
c:=Z
Gsym(q)
(gc)dµ(g), (7)
where the integration is over the symmetry group,
Gsym
, whose elements are
g
, with respect to the invariant measure,
µ
,
normalized so that
µ
(
Gsym
) = 1 , as described by Gazis et al. [
7
];
q
denotes the dependence of the value of integral (7) on the
relative orientations of
c
and
ˆ
c
. This integral reduces to a finite sum for the classes whose symmetry groups are finite, which are
all classes except isotropy and transverse isotropy. As shown by Gazis et al. [
7
], projection (7) ensures that a positive-definite
tensor is projected to another positive-definite tensor, as required by Hookean solids.
As shown by Moakher and Norris [9]and Bucataru and Slawinski [3]—in a fixed orientation of the rotation-symmetry axis
that coincides with the
x3
-axis of the coordinate system—the Frobenius-norm effective transversely isotropic tensor derived from
expression (7) has the form of tensor (3) with components given by
ˆ
cTI
1111 =1
83c1111 +3c2222 +2c1122 +4c1212, (8)
ˆ
cTI
1122 =1
8c1111 +c2222 +6c1122 4c1212, (9)
ˆ
cTI
1133 =1
2c1133 +c2233, (10)
ˆ
cTI
3333 =c3333 , (11)
ˆ
cTI
2323 =1
2c2323 +c1313, (12)
where
ci jk`
are the components of the generally anisotropic tensor. No analytic form of the operator-norm effective tensor is
known.
In general, the distance between the generally anisotropic tensor and its effective counterpart,
ceff
, expressed in the same
orientation of the coordinate system, is
d(q) =
cceff(q)
, (13)
where
q
denotes the orientation dependence. For the Frobenius case,
kkF
, where the norm is given by expression (5), average (7)
is tantamount to the orthogonal projection, and in view of that projection,
cˆ
c
and
ˆ
c
are normal to one another; hence, we can
write
d2
F(q) = kck2
Fkˆ
c(q)k2
F. (14)
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Danek ·Slawinski 3
No such simplification is possible for the operator norm due to the absence of the concept of orthogonality, and angle, in general,
since, herein, the operator norm is not an inner-product norm. Thus,
d2(q) =
c˜
c(q)
2, (15)
where
˜
c
is the operator-norm effective tensor. In the absence of analytic expressions, we compute the components of
˜
c
numerically,
which contributes to a difference in a computational difficulty between the Frobenius-norm and the operator-norm approaches.
In general—since expression (13) is orientation-dependent—to obtain the effective tensors, we must find the absolute
minima of expressions (14) and (15), which are the minima under all orientations of the rotation-symmetry axis. This is a highly
nonlinear problem, which we address with a global optimization method called particle-swarm optimization (PSO), discussed by
Poli et al. [10], and by invoking quaternions, as used by Kochetov and Slawinski [8]and Danek et al. [4].
For the Frobenius case, the search for the absolute minimum of expression (14) is achieved by expressing
c
in all orientations
of the coordinate system and, for each orientation, calculating
ˆ
c
whose components are stated in expressions (8)–(12). Since
kck
is invariant under coordinate transformations, it suffices to maximize
kˆ
ck
to minimize expression (14). This is a two-variable
problem, whose variables are the angles that define the orientation of the rotation-symmetry axis of a transversely isotropic
tensor.
Since no analytic form akin to expressions (8)–(12) is known for the operator effective tensor, we must search numerically
for the corresponding
˜
c
; this effective tensor must have a form of tensor (3), and be such that the maximum eigenvalue of the
difference between
c
and
˜
c
is minimized; we must perform minimization under all orientations of the rotation-symmetry axis.
This is a seven-variable problem, whose variables are the two angles that describe the orientation of the axis and the five elasticity
parameters of the corresponding effective tensor,
˜
c
. Furthermore, one must verify that each candidate for the effective tensor
is positive-definite, as required for a Hookean solid, since—unlike for the Frobenius-effective tensor—this requirement is not
intrinsically satisfied for the operator effective tensor, as discussed by Bos and Slawinski [2].
3 Numerical example
3.1 Effective tensors
To examine differences between the Frobenius norm and the operator norm, let us compare the transversely isotropic effective
tensors obtained using these norms. To do so, we use the generally anisotropic elasticity tensor obtained from seismic
measurements by Dewangan and Grechka [5],
c=
7.8195 3.4495 2.5667 p2(0.1374)p2(0.0558)p2(0.1239)
3.4495 8.1284 2.3589 p2(0.0812)p2(0.0735)p2(0.1692)
2.5667 2.3589 7.0908 p2(0.0092)p2(0.0286)p2(0.1655)
p2(0.1374)p2(0.0812)p2(0.0092)2(1.6636)2(0.0787)2(0.1053)
p2(0.0558)p2(0.0735)p2(0.0286)2(0.0787)2(2.0660)2(0.1517)
p2(0.1239)p2(0.1692)p2(0.1655)2(0.1053)2(0.1517)2(2.4270)
, (16)
whose entries of this matrix are the density-scaled elasticity parameters; their units are
km2/s2
. In other words, the Hookean
solid in question is completely described by expression (16). The corresponding standard deviations are
S=±
0.1656 0.1122 0.1216 p2(0.1176)p2(0.0774)p2(0.0741)
0.1122 0.1862 0.1551 p2(0.0797)p2(0.1137)p2(0.0832)
0.1216 0.1551 0.1439 p2(0.0856)p2(0.0662)p2(0.1010)
p2(0.1176)p2(0.0797)p2(0.0856)2(0.0714)2(0.0496)2(0.0542)
p2(0.0774)p2(0.1137)p2(0.0662)2(0.0496)2(0.0626)2(0.0621)
p2(0.0741)p2(0.0832)p2(0.1010)2(0.0542)2(0.0621)2(0.0802)
. (17)
As shown by Danek et al. [
4
], the effective tensor—in the Frobenius sense—is at the distance of 1
.
0733
km2/s2
from
tensor (16) , and is given by
ˆ
c=
8.0641 3.3720 2.4588 0 0 0
3.3720 8.0640 2.4588 0 0 0
2.4588 2.4588 7.0817 0 0 0
0 0 0 2(1.8625)0 0
0 0 0 0 2(1.8625)0
0 0 0 0 0 2(2.3460)
, (18)
with orientation determined by
q=±(0.993636 0.010573 i+0.027463 j+0.108726 k), (19)
where
q
is a unit quaternion,
q
=
a
+
bi
+
cj
+
dk
, where
a2
+
b2
+
c2
+
d2
=1 ,
i2
=
j2
=
k2
=
i j k
=
1 , and
a,b,c,d
are real
numbers.
Recall that a unit quaternions gives rise to a rotation in the space of purely imaginary quaternions,
p
, which are quaternions
whose
a
=0 . This is a consequence of the fact that
p7→ q p ¯
q
, where
¯
qabicjdk
is the conjugate of
q
, maps purely
imaginary quaternions to purely imaginary quaternions, which we can view as vectors relative to basis
{i j k}
. Also, since
kpk
=1 ,
it follows that
kqp¯
qk
=
kpk
, which means that it is a norm-preserving transformation. Thus, we can write
p0
=
q p ¯
q
, where
p0
is
protated by
a2+b2c2d22ad +2bc 2ac +2bd
2ad +2bc a2b2+c2d22ab +2cd
2ac +2bd 2ab +2c d a2b2c2+d2
, (20)
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Danek ·Slawinski 4
which is the rotation matrix corresponding to q.
The effective tensor—in the operator sense—is at the distance of 0.6022 km2/s2, and is given by
˜
c=
8.1154 3.1557 2.6736 0 0 0
3.1557 8.1154 2.6736 0 0 0
2.6736 2.673646 6.9239 0 0 0
0 0 0 2(1.8630)0 0
0 0 0 0 2(1.8630)0
0 0 0 0 0 2(2.4798)
, (21)
with orientation determined by
q=±(0.989244 +0.009249 i0.004980 j0.145900 k). (22)
3.2 Physical quantities
Let us consider differences in physical quantities resulting from the choice of a norm. Since the velocity of the
P
wave along
the rotation-symmetry axis is
pc3333
, we see that the estimates based on tensors (18) and (21) are 2
.
84
km/s
and 2
.
85
km/s
,
respectively; in the context of seismic measurements, this might be a negligible difference. Notably, this difference is smaller
than for the effective isotropic tensors, where—as shown by Bos and Slawinski [
2
]—the velocities are 2
.
71
km/s
and 2
.
76
km/s
,
respectively. The smaller difference might be due to a larger number of parameters: two-parameter isotropic model versus
seven-parameter transversely isotropic model. A choice of a model must involve a consideration of errors associated with
ci jk`
,
which we examine below.
For a transversely isotropic medium, the velocity of the
SH
wave exhibits an elliptical dependence with the semiaxis parallel
to the rotation-symmetry axis given by
pc3333
and the other semiaxis by
pc1212
. The ellipticity is symptomatic of the strength of
anisotropy. For tensor (18), (
c1212 c2323
)
/c2323
is 0
.
26 , and for tensor (21), it is 0
.
33 ; thus, even in the context of a limited
accuracy of seismic measurements, this might be a nonnegligible discrepancy between information provided by the two norms.
A comparison of effective tensors is not complete without the examination of their orientations. Converting expressions (19)
and (22) to the Euler angles—which, even though less convenient for computations than quaternions, might be easier to
visualize—we see that the orientations of their rotation-symmetry axes differ by
θ
=0
.
30
and
φ
=2
.
75
, where
θ
and
φ
are the azimuth and tilt respectively. Again, in the context of seismic measurements—where the axis orientation is interpreted as
the orientation of subsurface layers—this might be a negligible difference.
7.0 7.5 8.0 8.5 2.5 3.0 3.5
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 6.5 7.0 7.5 8.0 8.5 9.0
3.5 4.0 4.5 5.0
−6 −4 −2 0 2 4 −5 0 5 10
Figure 1:
Histograms of the five density-scaled elasticity parameters of the Frobenius effective transversely isotropic tensors and of the two Euler
angles describing the azimuth,
θ
, and tilt,
φ
, of their rotation-symmetry axes. These histograms are obtained from realizations of tensor (16)
subject to errors (17); black vertical lines correspond to the error-free case. [4]
Dolomites Research Notes on Approximation ISSN 2035-6803
Danek ·Slawinski 5
7.6 7.8 8.0 8.2 8.4 8.6 2.5 3.0 3.5 4.0
1.8 2.0 2.2 2.4 2.6 2.8 3.0 6.0 6.5 7.0 7.5 8.0
3.2 3.4 3.6 3.8 4.0 4.2 4.4
−10 −5 0 5 10 −10 −5 0 5 10 15
Figure 2:
Histograms of the five density-scaled elasticity parameters of the operator effective transversely isotropic tensors and of the two Euler
angles describing the azimuth,
θ
, and tilt,
φ
, of their rotation-symmetry axes. These histograms are obtained from realizations of tensor (16)
subject to errors (17); black vertical lines correspond to the error-free case.
3.3 Perturbations
To examine the effect of errors on the distribution of results, let us consider the perturbation method, whose results are illustrated
in Figures 1and 2. Using errors (17) and the Monte-Carlo method, we perturb tensor (16) a thousand times, and—using either
norm—find the corresponding effective tensor.
The computational effort required to find the operator effective tensors is significantly greater than to find the Frobenius
effective tensors. There are two main reasons for the increased effort.
First, as discussed in Section 2, not only the orientation—as is the case for the Frobenius norm—but also the elasticity
parameters are the subject of search. Thus, the dimensionality of the problem is increased.
Second, the maximum eigenvalue has to be computed for every solution candidate. To do so, we apply a modification of
the PSO algorithm used by Danek et al. [
4
]by incorporating the singular value decomposition (SVD) into the target function.
Fortunately, the well-known flexibility of PSO allows us to obtain results effectively enough to consider a perturbation analysis.
Examining Figures 1and 2, we see that histograms—in particular,
θ
and
φ
—obtained for the operator norm are wider.
This is a result of the optimization process, where only the maximum energy in directions of the eigenvectors is taken into
consideration; thus, global minima are found within wider basins of attraction. Also, unlike in Figure 1, histograms in Figure 2
exhibit a lack of alignment between the maxima of empirical distributions and results obtained for the error-free data. The
understanding of this shift requires further investigation.
4 Conclusions
The choice of norm has an impact on the resulting effective tensor. Differences between the operator-norm effective tensor and
the Frobenius-norm effective tensor are due to several reasons; notably, the operator norm takes into account the index symmetry
of the tensor, ci jk`=ck`i j , whereas the Frobenius norm does not.
In general, an optimization of a target function can be based on any valid criterion. Herein, both approaches can be used for
perturbation methods to evaluate qualitatively the effect of errors. However, until a rigorous statistical analysis is performed, such
a distinction between the norms remains qualitative. A quantitative statistical analysis can be done using maximum-likelihood
estimators, which require error-weighted norms. Such an approach is under investigation.
Be that as it may, it is important to recognize that, while tensor (16) is a mathematical analogy for physical properties of
a material measured by Dewangan and Grechka [
5
], tensors (18) and (21) are approximations of that tensor. Two distinct
mathematical approximations referring to the same physical object remind us of an obvious, yet commonly neglected, distinction
between the physical world and the realm of mathematics.
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Danek ·Slawinski 6
Acknowledgments
We wish to acknowledge discussions with, and contributions of, Len Bos and Misha Kochetov; the input of the former was
instrumental in our considering the operator norm; the input of the latter was important in addressing reviewer’s comments.
Also, we wish to acknowledge the graphic and editorial support of Elena Patarini and David Dalton, respectively.
This research was performed in the context of The Geomechanics Project supported by Husky Energy. Also, this research was
supported partially by the Natural Sciences and Engineering Research Council of Canada and by the Polish National Science
Center under contract No. UMO-2013/11/B/ST10/04742.
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Dolomites Research Notes on Approximation ISSN 2035-6803
... The slowness curves for tensor (8) and its isotropic counterpart circles discussed in Sections 5.3.1, 5.3.2 and 5.3.3, below, are shown in Figure 7. Isotropic tensors examined in this section are counterparts of this tensor. ...
... The F 36 distance between tensors (8) and (11) is 2.0535 , where we note that tensor (11) is the closest isotropic tensor according to the operator-not the F 36 -norm. Thus, in spite of similarities between the isotropic tensors, the distance between them is large in comparison to their distances to tensor (8). This is an illustration of abstractness of the concept of distances in the space of elasticity tensors. ...
... Using a numerical search based on a simple random walk through a solution space with the target function being a difference between the minimized F 21 distance and the maximized F 36 distance, an elasticity tensor is generated that is further from isotropy than tensor (8) according to the F 36 norm, but closer to isotropy than tensor (8) according to the F 21 norm. The search results in with its corresponding isotropic counterparts, ...
Article
Full-text available
It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor, it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the Frobenius 21-component norm and the operator norm on a general Hookean solid. We find that both Frobenius norms result in similar isotropic counterparts, and the operator norm results in a counterpart with a slightly larger discrepancy. The reason for this discrepancy is rooted in the very definition of that norm, which is distinct from the Frobenius norms and which consists of the largest eigenvalue of the elasticity tensor. If we constrain the elasticity tensor to values expected for modelling physical materials, the three norms result in similar isotropic counterparts of a generally anisotropic tensor. To examine this important case and without loss of generality, we illustrate the isotropic counterparts by commencing from a transversely isotropic tensor obtained from a generally anisotropic one. Also, together with the three norms, we consider the L 2 slowness-curve fit. Upon this study, we infer that-for modelling physical materials-the isotropic counterparts are quite similar to each other, at least, sufficiently so that-for values obtained from empirical studies, such as seismic measurements-the differences among norms are within the range of expected measurement errors.
... One can distinguish a group of techniques, which are based on the separation of an additive part of the elasticity tensor (e.g., by way projecting), possessing one or another type of symmetry [34][35][36][37]. In this case, the identification problem is reduced to determination of the symmetric part, which is the closest in some metric to a given tensor, as e.g., in [38][39][40][41][42][43][44][45]. Note that depending on the choice of the metric, the separated parts may have special properties. ...
... Nevertheless, optimization problems analogous to Problems 1-3, can be formulated in terms of quite arbitrary metric on 4 3  . Various ways to define such metric are considered, e.g., in [34,36,37,40]. In [36], the symmetric approximations of elasticity tensors are constructed with the use of different approaches, which are based on the Frobenius, Log-Euclidean and Riemannian distance functions. ...
... It was shown that the ⋅ -optimal approximation of a positive definite tensor, unlike the ⋅ 4 3  -optimal one, may be not positive definite or unique in some degenerate cases. The results for the evaluation of the ⋅ -and ⋅ 4 3  -optimal approximations in the transversely isotropic class and their comparative analysis are presented in [40]. ...
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