ArticlePDF Available

Uncertainty Analysis of Effective Elasticity Tensors Using Quaternion-based Global Optimization and Monte-Carlo Method

Authors:

Abstract and Figures

We consider the problem of representing a generally anisotropic elasticity tensor, which might be obtained from physical measurements, by a tensor belonging to a chosen material symmetry class, so-called ‘effective tensor'. Following previous works on the subject, we define this effective tensor as the solution of a global optimization problem for the Frobenius distance function. For all nontrivial symmetry classes, except isotropy, this problem is nonlinear, since it involves all orientations of the symmetry groups. We solve the problem using a metaheuristic method called particle-swarm optimization and employ quaternions to parametrize rotations in 3-space to improve computational efficiency. One advantage of this approach over previously used plot-guided local methods and exhaustive grid searches is that it allows us to solve a large number of instances of the problem in a reasonable time. As an application, we can use Monte-Carlo method to analyze the uncertainty of the orientation and elasticity parameters of the effective tensor resulting from the uncertainty of the given tensor, which may be caused, for example, by measurement errors.
Content may be subject to copyright.
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 1 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY
TENSORS USING QUATERNION-BASED GLOBAL
OPTIMIZATION AND MONTE-CARLO METHOD
by T. DANEK
(Department of Earth Sciences, Memorial University, St. John’s, Newfoundland,
A1B 3X5, Canada and Department of Geoinformatics and Applied Computer Science,
AGH–University of Science and Technology, Kraków, Poland)
M. KOCHETOV
(Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland,
A1C 5S7, Canada)
and
M. A. SLAWINSKI
(Department of Earth Sciences, Memorial University, St. John’s, Newfoundland,
A1B 3X5, Canada)
[Received 1 April 2012. Revise 26 October 2012. Accepted 9 January 2013]
Summary
We consider the problem of representing a generally anisotropic elasticity tensor, which might be
obtained from physical measurements, by a tensor belonging to a chosen material symmetry class,
so-called ‘effective tensor’. Following previous works on the subject, we define this effective
tensor as the solution of a global optimization problem for the Frobenius distance function.
For all nontrivial symmetry classes, except isotropy, this problem is nonlinear, since it involves
all orientations of the symmetry groups. We solve the problem using a metaheuristic method
called particle-swarm optimization and employ quaternions to parametrize rotations in 3-space
to improve computational efficiency. One advantage of this approach over previously used plot-
guided local methods and exhaustive grid searches is that it allows us to solve a large number
of instances of the problem in a reasonable time. As an application, we can use Monte-Carlo
method to analyze the uncertainty of the orientation and elasticity parameters of the effective
tensor resulting from the uncertainty of the given tensor, which may be caused, for example, by
measurement errors.
1. Introduction
A Hookean solid is defined by the constitutive relation with an elasticity tensor, cijk, which relates
linearly the stress, σij, and strain, εkl , tensors:
σij =
3
k,=1
cijkεkl ,i,j∈{1,2,3}.(1)
<mslawins@mun.ca>
Q. Jl Mech. Appl. Math, Vol. 0. No. 0 © The Author, 2013. Published by Oxford University Press;
all rights reserved. For Permissions, please email: journals.permissions@oup.com
Advance Access publication xxx 2013. doi:10.1093/qjmam/hbt004
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 2 1–20
2DANEK et al.
The elasticity tensor, c, is assumed to satisfy index symmetries, cijkl =cjikl =cklij, and a positivity
condition, which makes it possible to represent cbya6×6 positive-definite symmetric matrix,
which we denote by C, and rewrite Equation (1) in matrix form:
σ11
σ22
σ33
2σ23
2σ13
2σ12
=
c1111 c1122 c1133 2c1123 2c1113 2c1112
c1122 c2222 c2233 2c2223 2c2213 2c2212
c1133 c2233 c3333 2c3323 2c3313 2c3312
2c1123 2c2223 2c3323 2c2323 2c2313 2c2312
2c1113 2c2213 2c3313 2c2313 2c1313 2c1312
2c1112 2c2212 2c3312 2c2312 2c1312 2c1212
ε11
ε22
ε33
2ε23
2ε13
2ε12
,(2)
The factors of 2 and 2 ensure that the basis of the stress-tensor and strain-tensor spaces is the same,
namely,
100
000
000
,
000
010
000
,
000
000
001
,1
2
000
001
010
,1
2
001
000
100
,1
2
010
100
000
.
Such a formulation was investigated by Walpole (1), Rychlewski (2,3), and Cowin and Mehrabadi
(4,5). We refer to matrix (2) as the Kelvin notation in view of his work (6), p. 110. This notation
has been discussed in textbooks, notably, Chapman (7).
As shown by several researchers (8,9), tensor cbelongs to one of eight material-symmetry classes,
from general anisotropy to isotropy. Each class is characterized by its symmetry group, which are
the orthogonal transformations, g, that leave cinvariant:
c=gc.(3)
Expressing cand gin an orthonormal basis in R3, we can rewrite Equation (3)as
cmnnq =
3
ijkl=1
AmiAnk cijkl AT
jpAT
lq,(4)
where AO(3) is the matrix representing g.
Given a generally anisotropic tensor c, we wish to examine its relations to the seven nontrivial
symmetry classes. Such relations can be understood in terms of the concept of distance to a given
symmetry class, which was proposed by Gazis et al. (10), and allow us to infer information about
properties of the material represented by c, such as its layering or fractures. Following several papers,
notably (1114), we consider the Frobenius norm to find the closest tensors that belong to particular
symmetry classes. Such a tensor is referred to as the effective tensor of the given class.
Our choice of Frobenius norm, which is also referred to as Euclidean, is determined by its
simplicity and mathematical convenience. Indeed, the invariance of Frobenius norm under orthogonal
transformations leads to elegant formulas for projections of a generally anisotropic elasticity
tensor onto nontrivial symmetry classes with fixed orientations; this property allows us to use
expression (16), below. Another norm with this invariance property is the so-called log-Euclidean
norm, discussed by Norris (15) and defined as the Frobenius norm of the logarithm of elasticity tensor
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 3 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 3
regarded as a positive-definite matrix in the form of expression (2). This norm has an additional
advantage of having the same value for the stiffness and compliance tensors; note that tensor c,to
which we refer as the elasticity tensor, is also called the stiffness tensor and its inverse the compliance
tensor. Methods presented in this paper can be easily adapted to deal with the log-Euclidean norm
as illustrated by Kochetov and Slawinski (14).
If we assume an a priori orientation of the symmetry group, which implies the orientation of
the symmetry planes and axes of the effective tensor, then the resulting optimization problem is
linear, and the closest tensor is given by the orthogonal projection with respect to the Frobenius
inner product. If we do not assume an a priori orientation, then the distance between an elasticity
tensor and a given symmetry class is obtained by finding the orientation that minimizes the distance.
The necessity of performing a search under all orientations—for the nontrivial symmetries, except
isotropy—leads to a highly nonlinear optimization problem. Hence, typically, there are many local
minima, and local-optimization methods must be restricted to the vicinity of the global minimum to
avoid convergence to a local minimum instead of the global one. Kochetov and Slawinski (1214)
proposed methods for such a restriction based on visual examination of distance plots. However,
this examination is impractical if we wish to find the effective tensors for thousands of values of
c, which may be needed, for example, for statistical simulations. Exhaustive global-optimization
methods based on a grid search tend to dramatically increase computational cost if solutions are to
be found with high accuracy. In the present paper, we address this problem by applying a global
optimization method called particle-swarm optimization. We implement it using the quaternionic
parametrization of SO(3), which has many advantages over the Euler-angle parametrizations, as
discussed by Kochetov and Slawinski (12,14). As an application of this method, we analyze the
generally anisotropic tensor obtained by Dewangan and Grechka (16) from seismic measurements.
We find the effective tensors for all nontrivial symmetry classes and, by means of the Monte-
Carlo method, determine the uncertainty of their orientation and elasticity parameters resulting from
measurement errors.
We begin this paper by recalling the nontrivial symmetry classes of elasticity tensors. Then we
state the optimization problem for finding the effective tensor of each class—except for isotropy,
which allows analytic expression of the effective tensor, as shown by Voigt (17)—in the language
of quaternions and outline the particle-swarm method. We complete this paper by a numerical study
of a generally anisotropic tensor of Dewangan and Grechka (16).
2. Symmetries of elasticity tensors
The eight material symmetry classes of elasticity tensors are isotropy, cubic symmetry, transverse
isotropy, tetragonal symmetry, trigonal symmetry, orthotropic symmetry, monoclinic symmetry and
general anisotropy. They are characterized by their symmetry groups, which are subgroups of O(3)
and should be considered up to conjugation in O(3).
Specifically, the symmetry group of an elasticity tensor cis conjugate to one of the following eight
groups (see Bóna et al. (9)):
Giso =O(3),(5)
Gcubic =AO(3) |A(ei)ej,for any i,j∈{1,2,3},(6)
GTI =±I,±Rθ,e3,±Mv|θ∈[0,2π),vin the e1e2-plane,(7)
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 4 1–20
4DANEK et al.
Transversely isotropic
(1.07)
Isotropic
(2.14)
Cubic
(1.78)
Trigonal
(1.03)
Orthotropic
(0.76)
Monoclinic
(0.53)
Generally anisotropic
Tetragonal
(1.04)
Fig. 1 Order relation of symmetry classes of elasticity tensors. Arrows indicate subgroups in this partial
ordering. For instance, the symmetry group of monoclinic class is a subgroup of all nontrivial symmetry groups,
in particular, of both orthotropic and trigonal; however, orthotropic is not a subgroup of trigonal or vice-versa.
Numbers in parentheses are distances from tensor (22) to effective tensors stated in expressions (24)–(32)
Gtetra =±I,±Rπ
2,e3,±Rπ
2,e3,±Me1,±Me2,±Me3,±M(1,1,0),±M(1,1,0),(8)
Gtrig =±I,±R2π
3,e3,±R2π
3,e3,±Me1,±Mcos π
3,sin π
3,0,±Mcos 2π
3,sin 2π
3,0,(9)
Gortho =±I,±Me1,±Me2,±Me3,(10)
Gmono =±I,±Me3,(11)
Ganiso ={±I},(12)
where Rθ,eidenotes the rotation around eiby θand Mvdenotes the reflection about the plane whose
normal is v. The order relation among these groups is shown in Fig. 1. The vertices are connected by
a line if the symmetry group corresponding to the vertex below is contained in the symmetry group
corresponding to the vertex above.
Since cis an even-rank tensor, Ibelongs to each of its symmetry groups. Hence, if cis invariant
under A, it is also invariant under A. Thus, without loss of generality, we can consider only rotations
for transformations in expressions (5)–(11), which is tantamount to replacing O(3) by SO(3).
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 5 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 5
We now recall the expressions for the seven nontrivial symmetry classes using Kelvin notation.
In this notation, transformation AO(3) of cijkcorresponds to transformation of Cgiven by
C→ ˜
AC ˜
AT, where
˜
A=
A2
11 A2
12 A2
13 2A12A13 2A11 A13 2A11 A12
A2
21 A2
22 A2
23 2A22A23 2A21 A23 2A21A22
A2
31 A2
32 A2
33 2A32A33 2A31 A33 2A31A32
2A21A31 2A22 A32 2A23A33 A23A32 +A22A33 A23A31 +A21A33 A22A31 +A21 A32
2A11A31 2A12A32 2A13A33 A13 A32 +A12A33 A13 A31 +A11A33 A12 A31 +A11 A32
2A11A21 2A12A22 2A13A23 A13 A22 +A12A23 A13 A21 +A11A23 A12 A21 +A11 A22
(13)
is an orthogonal 6 ×6 matrix.
For instance, a monoclinic tensor has 13 linearly independent components. Since it exhibits a single
symmetry plane, the orientation of its natural coordinate system is described by two independent
parameters, such as two Euler angles. Notably, in a coordinate system whose x3-axis is normal to
the symmetry plane, a monoclinic tensor is
c1111 c1122 c1133 00
2c1112
c1122 c2222 c2233 00
2c2212
c1133 c2233 c3333 00
2c3312
0002c2323 2c2313 0
0002c2313 2c1313 0
2c1112 2c2212 2c3312 002c1212
.(14)
Remaining symmetry classes have fewer linearly independent components and a different pattern of
zero and nonzero entries, as shown in expressions (25)–(32), below. Except for the transverse isotropy
and isotropy, orientations of their natural coordinate systems are described by three independent
parameters. Transverse isotropy, which exhibits a single rotation-symmetry axis, is described by two
parameters. For an isotropic tensor,
c1111 c1111 2c2323 c1111 2c2323 000
c1111 2c2323 c1111 c1111 2c2323 000
c1111 2c2323 c1111 2c2323 c1111 000
0002c2323 00
00002c2323 0
000002c2323
,(15)
the arrangement of the zero and nonzero entries and the values of the latter remain the same for all
orientations of an orthonormal coordinate system, since each such system is a natural one.
3. Optimization problem
With the fixed orientation of the symmetry group, we can relate tensor cto its counterpart, csym,
which belongs to a particular symmetry class, sym, and can be viewed as the effective tensor of cin
that class, with the chosen orientation. Tensor csym is the orthogonal projection of c, in the sense of
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 6 1–20
6DANEK et al.
the Frobenius inner product, onto the linear space consisting of all tensors invariant under the group
Gsym, as discussed in Section 2. This projection is the average given by
csym := prsym (c)=
Gsym
(gc)dμ(g),(16)
where the integration is over gGsym, with respect to the invariant measure, μ, normalized so that
μ(Gsym)=1, as described by Gazis et al. (10). Integral (16) reduces to a finite sum for the classes
whose symmetry groups are finite, which are all classes except isotropy and transverse isotropy.
As shown in (10), projection (16) ensures that a positive-definite tensor is projected to another
positive-definite tensor, as required by Hookean solids.
The explicit expressions of csym for the monoclinic, orthotropic, tetragonal, transversely isotropic,
trigonal and cubic symmetries, in the coordinate system associated with the symmetry group of each
class as stated in Section 2, can be found in (18). We note that, for monoclinic and orthotropic classes,
the projection amounts to replacing certain entries of cby zeros, as exhibited—for monoclinic
class—by matrix (14).
Since, according to the definition of orthogonal projection, tensors cprsym (c)and prsym (c)
are normal to one another, we have
cprsym (c)
2=c2−prsym (c)2.(17)
Using the Kelvin notation, we can write the squared distance between cand csym as
d2
sym := C2−Csym 2.(18)
In all cases except isotropy, tensor prsym(c), and hence distance (18), depend on the chosen orientation
of the symmetry group. Thus the effective tensors considered so far are relative to the orientation
of the symmetry groups chosen a priori. To determine the effective tensor without assuming an a
priori orientation, we must minimize expression (18) under all possible orientations. We refer to the
resulting tensor as the absolute effective tensor of a given symmetry class.
Finding the minimum of expression (18) as a function of orientation is a nonlinear minimization
problem. Indeed, it follows from projection (16) that the effective tensor relative to the orientation of
Gsym that differs from the original orientation by rotation ASO(3) is given by ˜
Aprsym (˜
ATC˜
A)˜
AT,
and we have to perform the minimization of dsym under all ASO(3). We note that the global
minimum is attained for more than one matrix A. Indeed, if NSO(3) normalizes the group Gsym,
which means that NG
sym NT=Gsym, then Aand AN deliver the same effective tensor and hence
the same value of dsym.
To carry out computations for finding effective tensors, we parametrize SO(3) by quaternions,
which were introduced by W. R. Hamilton in the first half of the 19th century, and are described in
the context of rotations in such textbooks as Stillwell (19). We write a quaternion as
q=a+bi+cj+dk,
where i2=j2=k2=ijk=−1. We can view the quaternion as a scalar-vector pair, (a,[b,c,d])=:
(a,q), with abeing referred to as the real part and qas the imaginary part. The norm of a quaternion
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 7 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 7
is given by q2=q¯qqq =a2+q·q=a2+b2+c2+d2, where ¯q=(a,q) is the conjugate of
q. Unit quaternions, q=1, give rise to rotations in the space of purely imaginary quaternions, p,as
follows: p→ qp¯q. It is easy to verify that this mapping indeed sends purely imaginary quaternions
to purely imaginary quaternions and also preserves the norm. The corresponding orthogonal 3 ×3
matrix is given by
A(q)=
a2+b2c2d22ad +2bc 2ac +2bd
2ad +2bc a2b2+c2d22ab +2cd
2ac +2bd 2ab +2cd a2b2c2+d2
;(19)
this is the rotation by angle θ=2 arccos aabout the axis whose components are [b,c,d](relative to
basis {i,j,k}). The mapping q→ A(q) is not one-to-one, since ±qcorrespond to the same rotation:
(q)p(−¯q)=qp¯q. Following expression (13), ˜
A(q) gives rise to an orthogonal transformation
of the space of elasticity tensors, C→ ˜
A(q)TC˜
A(q). Substituting C(q)=˜
A(q)TC˜
A(q) for C
in expression (18) and using O(3)-invariance of the Frobenius norm, we obtain the following
formulation of our optimization problem:
d2
sym(q):= C2−prsym (C(q))2min .(20)
The solutions of this problem are the values of qthat result in the absolute minimum of the distance.
The corresponding C(q) is the absolute effective tensor, Csym
eff , expressed in the coordinate system
obtained from the original one by rotation A(q). The nonuniqueness of qdescribed above does
not result in nonuniqueness of the effective tensor, since the resulting matrices C(q) are merely
expressions of the same tensor in different coordinate systems. The expression of the effective
tensor in the original coordinate system, ˜
A(q)C˜
A(q)T, is the same for all such q, but, in general,
does not exhibit patterns illustrated by matrix (14).
We solve problem (20) by a particle-swarm optimization method (PSO). This method is used
as a global optimization strategy since it does not require any internal parameter tuning like other
powerful metaheuristic methods, such as a genetic algorithm, discussed by Donelli et al. (20). PSO
is a stochastic technique that simulates social behavior of animals searching for food. Every particle
is defined in an n-dimensional solution space. Particles are ‘aware’ of their current position, xi,of
their best individual position, pi, and of the best global position, pg, which is the best position of the
group. In our implementation, we use the ‘global best’ static topology, which means that the whole
swarm is a fully connected graph with no additional subdivisions, as discussed in (21).
A flow chart is shown in Fig. 2. A population array is initialized with random positions, pi, and
particle velocities, vi. Herein, the random positions are the real and imaginary parts of quaternions.
In the main computational loop, a forward problem is solved for a proposed rotation. Obtained
results are tested against the distance to a tensor exhibiting a desired symmetry and the pivalue
is updated if the xiposition results in a smaller distance. In subsequent iterations, a distance to the
selected symmetry class is minimized by testing pgagainst all values corresponding to the updated pi
solution. To obtain new positions and velocities, we use the canonical version of the update strategy
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 81–20
8DANEK et al.
Fig. 2 Flowchart describing the particle-swarm-optimization algorithm: xiand pirepresent, respectively, the
current and the best position of a single particle, pgdenotes the best position for the entire swarm, which herein
means the four parts of the quaternion defining the rotation resulting in the shortest distance to a given symmetry
formulated by Clerk and Kennedy (22),
viχvi+U(0,
1)(pixi)+U(0,
2)pgxi,
xixi+vi,
=1+2>4,
χ=2
2+24
,(21)
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 9 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 9
where Urepresents uniform distribution and is a componentwise multiplication. 1and 2
are constriction coefficients introduced to limit amplitudes of particle movements related to the
best individual and global positions, respectively. Commonly, is set to 4.1, which means that a
constant velocity damper, χ,is0.7298, and—after multiplication—Uis a random number between
0 and 2.05, if 1=2=2.05. Such a scheme guarantees convergence without particle-velocity
limitations, even though—to speed up computations—the velocity is usually limited to a maximum
value of xi, as discussed by Eberhart and Shi (23). In our implementation, since we use only unit
quaternions, we limit xito the closed interval of [−1,1], which allows us to explore effectively the
solution space but requires a tested quaternion to be normalized to unity after each iteration. In this
normalization—to prevent numerical instability—quaternions whose norms are close to zero are not
accepted.
4. Numerical example
4.1 General anisotropy
Let us consider a generally anisotropic elasticity tensor obtained by Dewangan and Grechka (16)
from seismic measurements,
C=
7.8195 3.4495 2.5667 2(0.1374)2(0.0558)2(0.1239)
3.4495 8.1284 2.3589 2(0.0812)2(0.0735)2(0.1692)
2.5667 2.3589 7.0908 2(0.0092)2(0.0286)2(0.1655)
2(0.1374)2(0.0812)2(0.0092)2(1.6636)2(0.0787)2(0.1053)
2(0.0558)2(0.0735)2(0.0286)2(0.0787)2(2.0660)2(0.1517)
2(0.1239)2(0.1692)2(0.1655)2(0.1053)2(0.1517)2(2.4270)
.
(22)
The entries of this matrix are the density-scaled elasticity parameters; their units are km2/s2.In
subsequent sections, we rotate tensor (22) to find a rotation that is a solution of problem (20); for
isotropy, no rotations are needed. Thus we obtain the shortest distances to each nontrivial symmetry
class and the corresponding effective tensors. Rotations are done using unit quaternions; PSO is used
as a global-optimization engine.
4.2 Monoclinic symmetry
For the monoclinic symmetry, a solution of (20), where Cis stated in (22), is
q(0.995432 0.012716 i+0.041023 j+0.085262 k),
which, in view of expression (19), corresponds to the rotation matrix given by
A=
0.982095 0.170789 0.079502
0.168703 0.985137 0.032311
0.083839 0.018320 0.996311
.(23)
The distance is dmono =0.529999 km2/s2, which as expected in view of Fig. 1and as confirmed by
the results below, is the shortest among all distances between tensor (22) and the seven nontrivial
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 10 1–20
10 DANEK et al.
symmetry classes. The corresponding effective monoclinic tensor is
Cmono
eff =
7.777106 3.416632 2.498631 0 0 2(0.026388)
3.416632 8.259775 2.422995 0 0 2(0.189542)
2.498631 2.422995 7.075589 0 0 2(0.215256)
0002(1.636567) 0 0
00002(2.091075) 0
2(0.026388) 2(0.189542) 2(0.215256) 0 0 2(2.392117)
.
(24)
Unlike in (14), herein, c2313 =0. Expression (24) is obtained by rotating the coordinate system about
the symmetry-plane normal, as discussed, for instance, by Slawinski (24). Note that the number of
parameters remains the same since, even though there are fewer nonzero entries, there is a parameter
describing the rotation about the normal.
4.3 Orthotropic symmetry
The distance to the orthotropic class is obtained by rotating tensor (22) using
q(0.814828 +0.006392 i+0.030504 j0.578865 k);
dortho =0.775189 km2/s2, and the corresponding effective tensor is
Cortho
eff =
8.376203 3.363446 2.487841 0 0 0
3.363446 7.774022 2.427617 0 0 0
2.487841 2.427617 7.080958 0 0 0
0002(2.078425) 0 0
000 02(1.649692) 0
000 0 02(2.332288)
.(25)
We note that the global minimum of distance is also attained with other rotations, as expected from
the normalizer of the group Gortho; see expression (10). They merely yield certain permutations of
the rows and columns of matrix (25). One could use the least deviation of the rotation matrix from
identity as the criterion for choosing the expression for the effective tensor.
4.4 Tetragonal symmetry
The distance to the tetragonal class is obtained by rotating tensor (22)by
q(0.998548 0.004725 i+0.031169 j0.043680 k);
dtetra =1.041555 km2/s2, which is greater that dortho and smaller than dTI (below), as expected in
view of Fig. 1. The corresponding effective tensor is
Ctetra
eff =
7.972472 3.462673 2.459470 0 0 0
3.462673 7.972472 2.459470 0 0 0
2.459470 2.459470 7.080814 0 0 0
0002(1.862168) 0 0
000 02(1.862168) 0
000 0 02(2.438777)
.(26)
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 11 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 11
Again, other rotations can be used to obtain the same value of dtetra. For example,
q(0.905875 +0.007554 i+0.030597 j0.422371 k)
yields
Ctetra
eff =
8.156348 3.278799 2.459469 0 0 0
3.278799 8.156348 2.459469 0 0 0
2.459469 2.459469 7.080820 0 0 0
0002(1.862168) 0 0
000 02(1.862168) 0
000 0 02(2.254901)
,(27)
which is the same tensor, Ctetra
eff , but expressed in a coordinate system that differs from the coordinate
system of (26) by the rotation by π/4 about its x3-axis.
4.5 Transverse isotropy
Distance to transverse isotropy is dTI =1.073258 km2/s2, which is only slightly higher than to
tetragonal symmetry. This distance is obtained by rotation with
q(0.993636 0.010573 i+0.027463 j+0.108726 k).
The effective tensor is
CTI
eff =
8.064069 3.372014 2.458775 0 0 0
3.372014 8.064069 2.458775 0 0 0
2.458775 2.458775 7.081722 0 0 0
0002(1.862519) 0 0
000 02(1.862519) 0
000 0 02(2.346028)
.(28)
Any rotation of the coordinate system about the rotation-symmetry axis results in the same expression
for tensor (28), since GTI =SO(2). This is analogous to isotropy, where expression (32), below,
remains unchanged under any rotation, since Giso =SO(3).
4.6 Trigonal symmetry
Distance to trigonal symmetry is dtrig =1.034372 km2/s2, obtained by
q(0.937228 0.010950 i+0.020294 j+0.347955 k).
We note that dortho <dtrig <dtetra, which is consistent with, but does not follow from, Fig. 1, due
to partial ordering: Gtrig is not a subgroup of Gortho or GTI. The effective tensor is
Ctrig
eff =
8.063584 3.369495 2.459688 0 2(0.073572) 0
3.369495 8.063584 2.459688 0 2(0.073572) 0
2.459688 2.459688 7.084078 0 0 0
0002(1.861663) 0 2(0.073572)
2(0.073572) 2(0.073572) 0 0 2(1.861663) 0
0002(0.073572) 0 2(2.347044)
.
(29)
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 12 1–20
12 DANEK et al.
Again, the component expression of the effective tensor is not unique. For example,
q(0.985617 +0.000711 i+0.023064 j0.167409 k)
yields
Ctrig
eff =
8.063587 3.369463 2.459703 0 2(0.073584) 0
3.369463 8.063587 2.459703 0 2(0.073584) 0
2.459703 2.459703 7.084070 0 0 0
0002(1.861654) 0 2(0.073584)
2(0.073584) 2(0.073584) 0 0 2(1.861654) 0
0002(0.073584) 0 2(2.347062)
,
(30)
which is another expression of tensor (29), up to a rotation by π/3 about the x3-axis.
4.7 Cubic symmetry
The distance to the cubic symmetry is 1.776056 km2/s2and results from
q(0.343847 +0.035674 i+0.007560 j0.938317 k).
The effective tensor is
Ccubic
eff =
7.799075 2.731962 2.731962 0 0 0
2.731962 7.799075 2.731962 0 0 0
2.731962 2.731962 7.799075 0 0 0
0002(1.992462) 0 0
000 02(1.992462) 0
000 0 02(1.992462)
.(31)
4.8 Isotropy
The distance to isotropy is 2.135352 km2/s2, which is the largest of all symmetry classes, as expected
from Fig. 1. The effective tensor, which is in the form of matrix (15), is
Ciso
eff =
7.366202 2.948401 2.948401 0 0 0
2.948401 7.366202 2.948401 0 0 0
2.948401 2.948401 7.366202 0 0 0
0002(2.208901) 0 0
000 02(2.208901) 0
000 0 02(2.208901)
.(32)
Unlike other effective tensors, it can be found analytically using
ciso
1111 =1
15 (3(c1111 +c2222 +c3333)+2(c1122 +c1133 +c2233)+4(c1212 +c1313 +c2323 ))
(33)
and
ciso
2323 =1
15 (c1111 +c2222 +c3333 (c1122 +c1133 +c2233)+3(c1212 +c1313 +c2323)) ,(34)
as shown by Voigt (17); no optimization is needed.
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 13 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 13
5. Error analysis
The entries of tensor (22) stated by Dewangan and Grechka (16) have the following standard
deviations:
S
0.1656 0.1122 0.1216 2(0.1176)2(0.0774)2(0.0741)
0.1122 0.1862 0.1551 2(0.0797)2(0.1137)2(0.0832)
0.1216 0.1551 0.1439 2(0.0856)2(0.0662)2(0.1010)
2(0.1176)2(0.0797)2(0.0856)2(0.0714)2(0.0496)2(0.0542)
2(0.0774)2(0.1137)2(0.0662)2(0.0496)2(0.0626)2(0.0621)
2(0.0741)2(0.0832)2(0.1010)2(0.0542)2(0.0621)2(0.0802)
.(35)
We denote these entries by Smn. Unlike matrix (22), matrix (35) does not represent components of
a tensor; it does not allow tensorial transformations.
5.1 Monte-Carlo method
An evaluation of uncertainty of results is necessary for proper interpretation, as discussed by
Eisenhart (25). In our case, it can, for example, prevent lowering of effective symmetry due to
errors, as discussed in (26).
The problem of uncertainty of symmetries of elasticity tensors and elasticity parameters was
examined by several researchers, notably by Guilleminot and Soize (27). Herein, we exemplify
error sensitivity of effective tensors by subjecting the entries of tensor (22) to normally distributed
errors with standard deviations given by (35).
Error propagation and application of stochastic methods are discussed in several textbooks,
notably, by Bevington (28). The Monte-Carlo method, which we apply herein, allows us to obtain
information about distribution and uncertainty of results. It is a numerical procedure based on repeated
calculations of a dependent quantity, which is the desired result, assuming that the independent
quantities are measured with known precision; also, distributions of these measurements must
be known or assumed. This method is used commonly if relations between input and results are
too complicated to use standard methods of error-propagation calculus, which can be exceedingly
laborious or even inapplicable, as discussed by Ku (29). In our case, the nonlinearity of the
distance-minimization problem renders analytical methods unusable.
To generate a statistical sample for error-propagation analysis, we replace each entry of matrix (22)
with N(Cmn,Smn ), where Ndenotes normal distribution. Since the covariances are not given by
Dewangan and Grechka (16), we assume that the errors corresponding to distinct entries of Care
independent. We repeat this process fifty-thousand times, each one independent from the others.
For each such realization of C, we obtain the effective monoclinic, orthotropic, tetragonal and
transversely isotropic tensors, which form the right-hand branch of Fig. 1, used commonly in
seismology. In presented examples, data generated by an original parallel C code were analyzed
using R environment for statistical computing, described in (30).
5.2 Orientation
First we examine the orientation of the effective tensors. As a reference, we use the orthonormal
coordinate system that corresponds to components (22), and which is the system used in the
experimental setup described by Dewangan and Grechka (16). Even though—for computational
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 14 1–20
14 DANEK et al.
−20 −10 0 10 20
−200 −100 0 100 200
−20
−10
0
10
Azimuth
−20 −10 0 10 20
−200 −100 0 100 200
−20
−10
0
10
Bank
Azimuth
Bank
Tilt
Tilt
Fig. 3 Orientations of the effective monoclinic, shown on the left, and orthotropic, shown on the right,
effective tensors. The axes denote the three Euler angles. Each black dot is the orientation of the effective tensor
corresponding to a realization of tensor (22) subject to errors (35). For picture clarity, the number of points is
limited to one thousand. Gray points are projections. The vertex of black lines is the orientation of the effective
tensor resulting from tensor (22), without errors, using the rotation matrix that is closest to identity
efficiency—the numerical operations are performed using quaternions, below we discuss the results
in terms of the Euler angles, which are more convenient for visualization.
Assigning Euler angles to rotations is an operation that requires particular care since it can be
done in 24 different ways (31). In this study, we use the convention described by Shoemake (32) and
apply rotation ZXZ with the rotating frame, which means that the azimuth of an effective tensor is
defined by rotation about the x3-axis, which is vertical, the tilt is defined by rotation about the x1-axis
and the bank by rotation about the new x3-axis, which, for a nonzero tilt, is not vertical. To avoid
redundancy, we limit the azimuth range to (π/2/2]and the tilt and bank ranges to (π, π ].
Let us consider Fig. 3. A continuous distribution of points along the entire range of the bank in
the diagram on the left is a consequence of the dependence of the monoclinic symmetry on the
orientation of the symmetry-plane normal only. This is described by the azimuth and tilt; rotating
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 15 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 15
−20 −10 0 10 20
−200 −100 0 100 200
−20
−10
0
10
−20 −10 0 10 20
−200 −100 0 100 200
−20
−10
0
10
Bank
Azimuth
Bank
Tilt
Tilt
Azimuth
Fig. 4 Orientations of the effective tetragonal, shown on the left, and effective transversely isotropic, shown
on the right, tensors. The axes denote the three Euler angles. Each black dot is the orientation of the effective
tensor corresponding to a realization of tensor (22) subject to errors (35). For picture clarity, the number of
points is limited to one thousand. Gray points are projections. The vertex of black lines is the orientation of the
effective tensor resulting from tensor (22), without errors, using the rotation matrix that is closest to identity
the coordinate system about the symmetry-plane normal results in a different expression of the same
effective tensor. A clustered distribution of points along the entire range of the bank in the diagram
on the right of Fig. 3is a consequence of four points of minimum with the same orientation of the
x3-axis, spaced at about π/2 from each other. Rotating the coordinate system by π/2 about its x3-axis
results in a different expression of the same effective tensor.
Let us consider Fig. 4. A clustered distribution of points along the entire range of the bank in the
diagram on the left is a consequence of eight points of minimum with the same orientation of the
x3-axis, spaced at about π/4 from each other; rotating the coordinate system by π/4 about its x3-
axis results in a different expression of the same effective tensor. Acontinuous distribution of points
along the entire range of the bank in the diagram on the right is a consequence of the dependence
of the transversely isotropic symmetry on the orientation of rotation-symmetry axis only, which is
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 16 1–20
16 DANEK et al.
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
Azimuth Tilt
Fig. 5 Density-scaled elasticity parameters and Euler angles of the effective transversely isotropic tensors
obtained from realizations of tensor (22) subject to errors (35). The error-free values of elasticity parameters,
c1111 =8.06, c1122 =3.37, c1133 =2.46, c3333 =7.08 and 2c2323 =3.73, are marked by solid lines. The
two last diagrams are the marginal distributions corresponding to projections on the right side of Fig. 4. They
describe the orientation of the symmetry axis of the effective tensor relative to the original coordinate system
of tensor (22). The elasticity parameters for each effective tensor are expressed in any coordinate system whose
x3-axis coincides with the rotation-symmetry axis
described by the azimuth and tilt. Any rotation of the coordinate system about the rotation-symmetry
axis results in the same expression, since the symmetry group of transverse isotropy is O(2).
5.3 Elasticity parameters
In Fig. 5, we present the empirical distributions of the elasticity parameters of the effective TI tensor,
along with the empirical distributions of the azimuth and tilt describing its orientation. Similar figures
can be obtained for other symmetry classes. It is important to note that the elasticity parameters of
each effective tensor in the sample are relative to its own orientation of the coordinate system. Hence,
it makes sense to compare the values of a chosen elasticity parameter across the sample only if the
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 17 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 17
−0.4 −0.4 0 −0.4 0 −1.0 0 0 0.4 −0.6 0
−0.6 0 0.40 −0.4 0 −0.6 0 −0.6 0
−0.10 0 −0.4 0 −0.4 0 −0.6 0
00.6 0 0.4 −0.6 0
0 0.8
−0.6 0
0
−0.6 0
Fig. 6 Differences between the elasticity parameters of realizations of tensor (22), subject to errors (35), and
the corresponding parameters of the effective transversely isotropic tensors. All tensors are expressed in the
original coordinate system of tensor (22)
orientations are close to each other. For instance, in the case of orthotropic and tetragonal symmetry
discussed above, one must select the orientations belonging to only one cluster—say, the closest to
the orientation of the original coordinate system of the experimental setup.
Alternatively, we can express all effective tensors in the original coordinate system. This
would mask the patterns illustrated by matrix (14), and would result, in general, in 21 nonzero
elasticity parameters. But this approach is convenient for estimating the likelihood that the
experimental data represents material of a given symmetry class. For instance, Fig. 6shows,
for each of the 21 elasticity parameters, the empirical distribution of the difference between
tensor (22) and the effective TI tensor. Judging from this figure, the experimental data obtained
by Dewangan and Grechka (16) is not very likely to correspond to a transversely isotropic Hookean
solid.
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 181–20
18 DANEK et al.
6. Discussions and conclusions
Having obtained all effective tensors, we are able to represent generally anisotropic tensors, such as
the one described by Dewangan and Grechka (16), by their symmetric counterparts and to examine
quantitatively their mathematical properties, from which we can infer mechanical properties of
materials represented by Hookean solids.
In summary, the Frobenius distances from the generally anisotropic tensor, stated in
expression (22), to the seven effective tensors, stated in expressions (24)–(32), are dmono =
0.529999,dortho =0.775189,dtrig =1.034372,dtetra =1.041555,dTI =1.073258,dcubic =
1.776056,diso =2.135352, which agrees with the partial ordering stated in Fig. 1: the greater
the subgroup, the greater the distance; hence, isotropy, which contains all subgroups, is furthest
from the generally anisotropic tensor. To avoid misunderstandings, we note that the sum of distances
between, say, the generally anisotropic tensor and its transversely isotropic effective tensor and from
that transversely isotropic to isotropic, namely: 1.073258 +1.846038 =2.919296, is greater than
between the generally anisotropic and isotropic tensors, which is 2.135352; this is a consequence of
the triangle inequality.
The values of distances are also in agreement with Kochetov and Slawinski (13), who used
their plot-guided local search to obtain dmono,dortho and dTI . Notably, similarity of orientations
of the x3-axes for the natural coordinates of effective monoclinic, orthotropic and transversely
isotropic tensors confirms the hypothesis of Kochetov and Slawinski (13), according to which finding
effective orthotropic tensors by minimizing the distance function can be guided by orientations
of monoclinic and transversely isotropic tensors, whose distance function can be plotted on
a sphere, since their orientation depends on two parameters only. The orientations of the x3-
axes for tensors (24), (25) and (28) are defined by vectors whose expressions in the coordinate
system of tensor (22) are [−0.08384,0.018320,0.996311],[−0.057136,0.024891,0.998056]
and [−0.056867,0.015055,0.998268], which are the third rows of the corresponding rotation
matrices. In view of the magnitudes and similarities of the third components of these vectors,
the differences in the former two components are not significant enough to be noticeable in a
three-dimensional plot.
Prior to the present work, the process of finding an effective tensor was based on human-guided
local optimization methods or on a grid search. The presented quaternion-based particle-swarm
optimization allows us to avoid human involvement and provides us with a fast and stable numerical
solver, which can be used for studies requiring simulations based on thousands of solutions.
We illustrate the usefulness of the presented method by implementing the Monte-Carlo error-
propagation analysis for the effective elasticity tensors of the monoclinic, orthotropic, tetragonal
and transversely isotropic symmetries associated with the generally anisotropic tensor (22). Such
analysis allows us to consider the range of possible interpretation of an elasticity tensor obtained from
measurements. In particular, we can consider a preference of a particular model over other models
using statistical methods—a topic for future work. We note that the comparison of the distance to
a given effective tensor and the norm of errors, which for matrix (35)isS=0.78434, is not
sufficient for this purpose because it favors lower symmetry over higher symmetry; for example, for
all generally anisotropic Hookean solids, the distance to a monoclinic effective tensor is less than
for other symmetries. Also, examination of the norm of errors ignores errors for individual elasticity
parameters.
Furthermore, this work gives insights into the very concept of a Hookean solid, in particular, and
constitutive relations, in general, which refer to mathematical entities defined by equations. Relations
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 19 1–20
UNCERTAINTY ANALYSIS OF EFFECTIVE ELASTICITY TENSORS 19
between real materials and such abstract entities must be mediated by measurements, which are
subject to errors, and a theory, which is subject to limitations. Presented results allow us to view a
Hookean solid not as a specific, albeit unknown, set of parameters with their associated uncertainties,
but as nondeterministic random variables whose solutions are their probability distributions, which is
a stochastic approach. This approach is particularly applicable in the context of ill-posed problems,
commonly encountered in seismology, since it allows us to make explicit assumptions about the
model, such as its exhibiting a particular symmetry.
Acknowledgments
The error analysis in this work was inspired by collaborations and discussions with the late Albert
Tarantola. Also, Scott Greenhalgh and Daniel Peter played important roles in our initial numerical
examinations of ideas presented in this paper. The authors acknowledge the editorial help of David
Dalton and graphic design by Elena Patarini. Also, the authors acknowledge the editorial guidance
of Paul Martin and detailed comments of the anonymous reviewer. T.D. received funding from
the Atlantic Innovation Fund and the Research and Development Corporation of Newfoundland
and Labrador through the High Performance Computing for Geophysical Applications Project.
M.K.’s and M.S.’s research was partially supported by the Discovery Grant of The Natural Sciences
and Engineering Research Council of Canada. This research was performed in the context of The
Geomechanics Project supported by Husky Energy.
References
1. L. J. Walpole, Fourth-rank tensors of the thirty-two crystal classes: multiplication tables. Proc.
R. Soc. A391 (1984) 149–179.
2. J. Rychlewski, On Hooke’s law. PMM J. Appl. Math. Mech. 48 (1984) 303–314.
3. J. Rychlewski, Unconventional approach to linear elasticity. Arch. Mech. 47 (1995) 149–171.
4. S. C. Cowin and M. M. Mehrabadi, On the structure of the linear anisotropic elastic symmetries.
J. Mech. Phys. Solids 40 (1992) 1459–1471.
5. S. C. Cowin and M. M. Mehrabadi, On the identification of material symmetry for anisotropic
elastic-materials. Q. J. Mech. Appl. Math. 40 (1987) 451–476.
6. W. Thomson (Lord Kelvin), Elements of a mathematical theory of elasticity. Phil. Trans. R. Soc.
146 (1856) 481–498.
7. C. H. Chapman. Fundamentals of Seismic Wave Propagation (Cambridge University Press,
Cambridge 2004).
8. S. Forte and M. Vianello, Symmetry classes for elasticity tensors. J. Elast. 43 (1996) 81–108.
9. A. Bóna, I. Bucataru and M. A. Slawinski, Material symmetries of elasticity tensors. Q. J. Mech.
Appl. Math. 57 (2004) 583–598.
10. D. C. Gazis, I. Tadjbakhsh and R. A. Toupin, The elastic tensor of given symmetry nearest to an
anisotropic elastic tensor. Acta Cryst. 16 (1963) 917–922.
11. A. N. Norris, The isotropic material closest to a given anisotropic material. J. Mech. Mater.
Struct. 1(2006) 223–238.
12. M. Kochetov and M. A. Slawinski, On obtaining effective orthotropic elasticity tensors. Q. J.
Mech. Appl. Math. 62 (2009) 149–166.
13. M. Kochetov and M. A. Slawinski, Estimating effective elasticity tensors from Christoffel
equations. Geophys. 74 (2009) WB67–WB73.
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
[15:13 13/3/2013 hbt004.tex] QJMAM: The Quarterly Journal of Mechanics & Applied Mathematics Page: 20 1–20
20 DANEK et al.
14. M. Kochetov and M. A. Slawinski, On obtaining effective transversely isotropic elasticity
tensors. J. Elast. 94 (2009) 1–13.
15. A. N. Norris, Elastic moduli approximation of higher symmetry for the acoustical properties of
an anisotropic material. J. Acoust. Soc. Amer. 119 (2006) 2114–2121.
16. P. Dewangan and V. Grechka, Inversion of multicomponent, multiazimuth walkaway VSP data
for the stiffness tensor. Geophys. 68 (2003) 1022–1031.
17. W. Voigt, Lehrbuch der Kristallphysics (Teubner, Leipzig 1910).
18. Ç. Diner, M. Kochetov and M. A. Slawinski, On choosing effective symmetry classes for
elasticity tensors. Q. J. Mech. Appl. Math. 64 (2011) 57–74.
19. J. Stillwell, Naive Lie Theory (Springer, Berlin 2008).
20. M. Donelli, G. Franceschini, A. Martini and A. Massa,An integrated multiscaling strategy based
on a particle swarm algorithm for inverse scattering problems. IEEE Trans. Geosci. Remote
Sensing 44 (2006) 298–312.
21. R. Poli, J. Kennedy and T. Blackwell, Particle swarm optimization. Swarm Intelligence 1(2007)
33–57.
22. M. Clerc and J. Kennedy, The particle swarm – explosion, stability, and convergence in a
multidimensional complex space. IEEE Trans. Evolutionary Computation 6(2002) 58–73.
23. R. C. Eberhart and Y. Shi. Proceedings of Congress on Evolutionary Computation (IEEE, La
Jolla, CA, USA 2000) 84–88.
24. M. A. Slawinski. Waves and Rays in Elastic Continua (World Scientific, Singapore 2010).
25. C. Eisenhart, Expression of the uncertainties of final results. Science 160 (1968) 1201–1204.
26. R. J. Arts, K. Helbig and P. N. J. Rasolofosaon, General anisotropic elastic tensor in rocks:
Approximation, invariants, and particular directions. 1991 SEG Annual Meeting (SEG,
Houston, TX, USA 1991) 1534–1537.
27. J. Guilleminot and C. Soize, A stochastic model for elasticity tensors with uncertain material
symmetries. Int. J. Solids Struct. 47 (2010) 3121–3130.
28. Ph. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill,
New-York 1969).
29. H. Ku, Notes on the use of propagation of error formulas. J. Res. Nat. Bureau Stand. 70C (1966)
263–273.
30. R Development Core Team, R: A Language and Environment for Statistical Computing (R
Foundation for Statistical Computing, Vienna 2012).
31. J. J. Craig, Introduction to Robotics: Mechanics and Control, 3rd edn. (Pearson/Prentice Hall,
Upper Saddle River 2005).
32. K. Shoemake, Euler angle conversion. Graphics Gems IV (ed. P. S. Heckbert; AP Professional,
San Diego, CA, USA 1994) 222–229.
at Memorial University of Newfoundland on March 15, 2013http://qjmam.oxfordjournals.org/Downloaded from
... where σ i j , k and c i jk are the stress, strain and elasticity tensors, respectively. The components of the elasticity tensor can be written-in Kelvin's, as opposed to Voigt's, notation (e.g., Chapman [15])-as a symmetric second-rank tensor in 6 , ...
... To examine the closeness between elasticity tensors, as discussed by Bos and Slawinski [3] and by Danek et al. [5,6], we consider several norms. ...
... which are the estimates of the standard deviations corresponding to each component of expression (6), in the coordinate system of data acquisition, since these values do not constitute components of a tensor. Either figure results from ten thousand repetitions. ...
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.
... Each of these tensors was the subject of separate PSO optimization, and the distributions of rotated tensor entries were obtained. The results obtained were compared with solutions of the method based on the Frobenius distances (Danek et al. 2013). ...
... For all symmetry groups except isotropy, the operator of the projection pr sym (c) as well as obtained distance (9), are dependent on the orientation of the symmetry group. To find the effective tensor without a priori assumption of orientation, one should minimize the squared distance under all orientations, and the tensor obtained in the result is referred to as an absolute effective (Danek et al. 2013). The problem of the minimization of Equation (9) as a function of orientation is nonlinear. ...
... The procedure described by several authors (i.e. Kochetov & Slawinski 2009, Danek et al. 2013 includes the selection of a particular symmetry class and then the optimization of the target function, which is a Frobenius norm (14), over different orientations of the coordinate system. This results in computational form of Equation (13): ...
... The Monte Carlo probability statistical method was used by Liu Qinjie et al. to analyze the uncertainty of the physical and mechanical parameters of a rock mass in a mining area (Liu and Yang 2016). Through the Monte-Carlo method, Danek T et al. analyzed the uncertainty of the orientation and the elasticity parameters of the effective tenor resulting from the uncertainty of the given tensor (Danek et al. 2013). Sheng Yanan et al. obtained the probability distribution characteristics of the formation collapse pressure and formation fracture pressure by an uncertainty analysis of the geomechanical parameters based on the Monte Carlo simulation method (Sheng et al. 2017). ...
... Because of the large variability and discreteness of the mechanical parameters in the same formation section, the uncertainty of formation mechanical parameters should be considered when calculating the in situ stress of the strata with a large dip angle, and the influence of the dip angle should not be neglected. In this paper, the Monte Carlo method (Danek et al. 2013) is used to analyze the uncertainty of formation mechanics parameters and in situ stress as following steps: ...
Article
Full-text available
In the development process of oil and gas field, the dip angles generally exist in deep strata, and the variability and dispersion in the mechanics parameters of the formation make the computation model of in situ stress for a gentle structure no longer applicable. In light of the fact that the dip angle was large and the geological structure was complex in Sichuan basin and Xinjiang basin, uncertainty of mechanical parameters and tectonic stress coefficients of the formation were analyzed by the Monte Carlo method based on acoustic logging data and stratigraphic dipmeter log data. Taking dip angle and dip direction into account, the uncertainty calculation method for in situ stress in deep inclined strata was proposed. Multi-factor analysis was used to analyze the sensitivity of various factors affecting the in situ stress in deep inclined strata. The results show that, the horizontal in situ stress calculation model adopted in this paper is more suitable for structurally intense areas. The horizontal stress confidence interval is more significant for oil and gas field engineering than single principal stress value. For PY gas field and TLM oil field with different geological conditions and rocks mechanical parameters of formation, the dip angle and effective stress coefficient are the most sensitive to the horizontal in situ stress.
... Without loss of generality, in this paper, we assume that the nonlinear convex optimization (1) has at least an optimal solution. The quaternion-valued optimization has been broadly applied in engineering control and scientific applications, such as picture recovery [1] [2] and color recognition [3], the computer-aided diagnosis (CAD) [4], the modeling of threedimensional (3-D) wind signa [5], automatic control [6], global exponential stability analysis [7] [8], multistability problem analysis [9] and optimization [10] [11]. Compared with complex domain and real domain, the quaternion field support a complete representation and essential approach to maintain the original physical characteristics, which taking the inner relationship into account. ...
... The AI cognitive modelling or neural network is an interconnected group of neurons that use a computational or mathematical model for information processing depending on a conjunctive method to computation. It has been widely used for predictive modeling [5] [11], imagine processing [3] [2] [14], engineering control [6] [7] [10] [19] [20] and clinical medicine [4] [12] via the flow of signals through the net connections. Owning to the seminal work of Tank and Hopfield [21], who presented a recurrent neural network to solve linear optimization problems in 1986, many researchers are motivated to exploit other neural networks for optimization [22]- [25]. ...
Article
This paper proposes a quaternion-valued one-layer recurrent neural network approach to resolve constrained convex function optimization problems with quaternion variables. Leveraging the novel generalized Hamilton-real (GHR) calculus, the quaternion gradient-based optimization techniques are proposed to derive the optimization algorithms in the quaternion field directly rather than the methods of decomposing the optimization problems into the complex domain or the real domain. Via chain rules and Lyapunov theorem, the rigorous analysis shows that the deliberately designed quaternion-valued one-layer recurrent neural network stabilizes the system dynamics while the states reach the feasible region in finite time and converges to the optimal solution of the considered constrained convex optimization problems finally. Numerical simulations verify the theoretical results.
... 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. ...
... There may exist multiple local and global minimums, so that the direct use of determined numerical methods is obstructed. However, for the majority of practical applications, there is no need for an exact approximation of the solutions to a global minimum, so that the optimization problems can be solved by using the heuristic algorithms, e.g., the particle swarm optimization method [71] employed in [39,41]. It seems plausible to use as a criterion of assigning a material by its properties to one or another class a small value of the residual caused by the related approximation in its constitutive equation. ...
Article
Full-text available
The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of "elasticity class" the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.
... Integral (2) reduces to a finite sum for the classes whose symmetry groups are finite, which are all classes in Fig. 1, except isotropy and transverse isotropy. The Gazis et al. [8] approach is reviewed and extended by Danek et al. [6,7] in the context of random errors. Therein, elasticity tensors are not constrained to the same-or even different but known-orientation of the coordinate system. ...
... , n, by 1 2 to find that, as expected, the commutator is multiplied by 1 4 . To quantify the strength of anisotropy, we invoke the concept of distance in the space of elasticity tensors (Danek et al. [6,7], Kochetov and Slawinski [9,10]). In particular, we consider the closest isotropic tensor-according to the Frobenius norm-as formulated by Voigt [14]. ...
Article
Full-text available
We show that, in general, the translational average over a spatial variable---discussed by Backus \cite{backus}, and referred to as the equivalent-medium average---and the rotational average over a symmetry group at a point---discussed by Gazis et al. \cite{gazis}, and referred to as the effective-medium average---do not commute. However, they do commute in special cases of a given symmetry class, which correspond to particular relations among the elasticity parameters. We also examine the extent of this noncommutativity, and show that it is a function of the strength of anisotropy. Surprisingly, a perturbation of the elasticity parameters about a point of weak anisotropy results in the commutator of the two types of averaging being of the order of the {\it square} of this perturbation. Thus, these averages nearly commute in the case of weak anisotropy, which is of interest in such disciplines as quantitative seismology.
... Many researchers have made efforts to analyze the uncertainty of the institution (Lee 1991;Kang and Bai 2013;Liu et al. 2015a, b). Scholars are devoted to analyzing errors in the length and clearance of mechanical joints of planar mechanisms (Liu et al. 2017a, b;Danek et al. 2013;Wu et al. 2019). Todorovi and Zelenovi (1980) has developed a reliability distribution method that can be applied to mechanical engineering systems. ...
Article
Full-text available
Considering manufacturing errors and installation errors in the automatic battery replacement system, it is necessary to analyze the uncertainty of the automatic battery replacement system to improve the efficiency and accuracy of the system. There are a few universal approaches for uncertainty analysis, such as interval analysis, probability analysis, and combination approach. This paper uses grey number theory, a relatively new method, to analyze. The analysis on the coordinates of the support points finds that the grey number theory can output a range smaller than that of the other three methods, and its results are in line with the actual situation. The grey number theory is used to analyze the coordinate position change of the support point of the automatic battery replacement device with the input error change and the driving rod angle change. The bar tolerance and error circle (installation error) existing in the system are also taken into consideration. Comparing the results produced by the case where error circle is considered and that is not, it can be seen that the influence of the error circle on the results is relatively insignificant compared with that of rod tolerance, but the influence cannot be ignored for structures requiring high precision. Therefore, the analysis of the input error percentage can yield a reasonable control range of the input error based on the grey number theory, which not only ensures the system is safe and reliable, but also is cost-efficient.
... A term of entropy and its properties are associated in engineering mostly with Maximum Entropy Principle, 7,17,18 which does not demand direct computations of its values, while its common application with the stochastic FEM [19][20][21] in various implementations is still rare. Most frequently probabilistic entropy is employed for stochastic reliability assessment 22 or for determination of probabilistic sensitivity 8,23 and still remains applicable in certain non-engineering studies. 2 Homogenization problem is one of the areas where probabilistic entropy computations can be interesting 20 because of many completely different sources of uncertainty and may be associated with an application of the Monte-Carlo simulation, 24 XFEM computer analysis, 25 or some other concurrent stochastic techniques. 1,4,20 2 | PROBABILISTIC ENTROPY CALCULATIONS It is known that Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy known from thermodynamics and has been introduced to generalize standard statistical mechanics. ...
Article
This work concerns an application of the Tsallis entropy to homogenization problem of the fiber-reinforced and also of the particle-filled composites with random material and geometrical characteristics. Calculation of the effective material parameters is done with two alternative homogenization methods – the first is based upon the deformation energy of the Representative Volume Element (RVE) subjected to the few specific deformations while the second uses explicitly the so-called homogenization functions determined under periodic boundary conditions imposed on this RVE. Probabilistic homogenization is made with the use of three concurrent non-deterministic methods, namely Monte-Carlo simulation, iterative generalized stochastic perturbation technique as well as the semi-analytical approach. The last two approaches are based on the Least Squares Method with polynomial basis of the statistically optimized order – this basis serves for further differentiation in the tenth order stochastic perturbation technique, while semi-analytical method uses it in probabilistic integrals. These three approaches are implemented all as the extensions of the traditional Finite Element Method (FEM) with contrastively different mesh sizes and they serve in computations of Tsallis entropies of the homogenized tensor components as the functions of input coefficient of variation.
Article
Both the distribution and orientation of anisotropic materials have great effects on the performance of civil and mechanical structures. Taking the maximum structural stiffness as the design objective, a concurrent method for determining the optimal distribution and orientation of orthotropic materials is proposed in this article. The method for determining material orientation theoretically overcomes the ‘repeated global minimum’ phenomenon, which often occurs in strain-based and stress-based methods, when the material is shear ‘strong’. Then, the concurrent optimization methodology for material distribution and orientation is developed by introducing an independent discrete topology variable and continuous material orientation angle. The optimized designs of material distribution and orientation are obtained concurrently by the gradient-based bi-directional evolutionary structural optimization method and the proposed material orientation optimization method. Numerical two- and three-dimensional examples are presented to demonstrate the effectiveness of the proposed concurrent optimization algorithm.
Article
Tilted orthorhombic (TOR) models are typical for dipping anisotropic layers, such as fractured shales, and can also be due to nonhydrostatic stress fields. Velocity analysis for TOR media, however, is complicated by the large number of independent parameters. Using multicomponent wide-azimuth reflection data, we develop stacking-velocity tomography to estimate the interval parameters of TOR media composed of homogeneous layers separated by plane dipping interfaces. The normal-moveout (NMO) ellipses, zero-offset traveltimes, and reflection time slopes of P-waves and split S-waves (S1 and S2) are used to invert for the interval TOR parameters including the orientation of the symmetry planes. We show that the inversion can be facilitated by assuming that the reflector coincides with one of the symmetry planes, which is a common geologic constraint often employed for tilted transversely isotropic media. This constraint makes the inversion for a single TOR layer feasible even when the initial model is purely isotropic. If the dip plane is also aligned with one of the symmetry planes, we show that the inverse problem for P-, S1-, and S2-waves can be solved analytically. When only P-wave data are available, parameter estimation requires combining NMO ellipses from a horizontal and dipping interface. Because of the increase in the number of independent measurements for layered TOR media, constraining the reflector orientation is required only for the subsurface layer. However, the inversion results generally deteriorate with depth because of error accumulation. Using tests on synthetic data, we demonstrate that additional information such as knowledge of the vertical velocities (which may be available from check shots or well logs) and the constraint on the reflector orientation can significantly improve the accuracy and stability of interval parameter estimation.
Article
Full-text available
We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular sym-metry that corresponds to measurable traveltime and polar-ization quantities. These quantities — the wavefront-slow-ness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization prob-lem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of partic-ular symmetry without assuming its orientation is challeng-ing. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we dis-tinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.
Book
Full-text available
The present book — which is the second, and significantly extended, edition of the textbook originally published by Elsevier Science — emphasizes the interdependence of mathematical formulation and physical meaning in the description of seismic phenomena. Herein, we use aspects of continuum mechanics, wave theory and ray theory to explain phenomena resulting from the propagation of seismic waves. The book is divided into three main sections: Elastic Continua, Waves and Rays and Variational Formulation of Rays. There is also a fourth part, which consists of appendices. In Elastic Continua, we use continuum mechanics to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such a material. In Waves and Rays, we use these equations to identify the types of body waves propagating in elastic continua as well as to express their velocities and displacements in terms of the properties of these continua. To solve the equations of motion in anisotropic inhomogeneous continua, we invoke the concept of a ray. In Variational Formulation of Rays, we show that, in elastic continua, a ray is tantamount to a trajectory along which a seismic signal propagates in accordance with the variational principle of stationary traveltime. Consequently, many seismic problems in elastic continua can be conveniently formulated and solved using the calculus of variations. In the Appendices, we describe two mathematical concepts that are used in the book; namely, homogeneity of a function and Legendre's transformation. This section also contains a list of symbols.
Article
Full-text available
We formulate a method of representing a generally anisotropic elasticity tensor by an elasticity tensor exhibiting a material symmetry: an effective tensor. The method for choosing the effective tensor is based on examining the features of the plot of the monoclinic-distance function of a given tensor, choosing an appropriate symmetry class, and then finding the closest tensor in that class. The concept of the effective tensor is not tantamount to the closest tensor since one always obtains a closer approximation using a monoclinic tensor than a tensor of any other nontrivial symmetry. Hence, we use qualitative features of the plot of the monoclinic-distance function to choose an effective symmetry class within which the closest tensor can be computed.
Article
Full-text available
Particle swarm optimization (PSO) has undergone many changes since its introduction in 1995. As researchers have learned about the technique, they have derived new versions, developed new applications, and published theoretical studies of the effects of the various parameters and aspects of the algorithm. This paper comprises a snapshot of particle swarming from the authors’ perspective, including variations in the algorithm, current and ongoing research, applications and open problems.
Article
Clear statements of the uncertainties of reported values are needed for their critical evaluation.
Article
A concise overview of a series of papers following [the author, J. Appl. Math. Mech. 48, 303-314 (1935; Zbl 0581.73015); translation from Prikl. Mat. Mekh. 48, 420-435 (1984)] has been presented; their principal feature consists in spectral decomposition of the stiffness and compliance tensors: the stiffness tensor is determined by six stiffness moduli and six mutually orthogonal stress-strain states, called proper elastic states. Numerous consequences of such a description have been analyzed.