# Generalized plane strain embedded in three-dimensional anisotropic
elasticity

Markus Lazar ${}^{\text{a,}}$111E-mail address<EMAIL_ADDRESS>, Helmut
O.K. Kirchner ${}^{\text{b,}}$222E-mail address<EMAIL_ADDRESS>

${}^{\text{a}}$ Karlsruhe Institute of Technology (KIT),
Institute of Engineering Mechanics,
D-76131 Karlsruhe, Germany
${}^{\text{b}}$ INM - Leibniz Institute for New Materials,
D-66123 Saarbrücken, Germany

###### Abstract

The theory of anisotropic generalized plane strain of line forces and
dislocation lines is embedded in three-dimensional elasticity of point forces
and dislocation densities. Embedding in real space is achieved by slicing in
reciprocal space using the projection-slice theorem.

Keywords: anisotropic elasticity; dislocations; line forces; projection-slice
theorem

##  1 Introduction

The theory of generalized plane strain, which considers two-dimensional (2D)
elastic fields caused by dislocations and line forces, has found its final
form in the so-called “6-dimensional integral theory” developed by Barnett and
Lothe [7] and codified by Bacon et al. [5] and Lothe [18]. The formalism
connects with previous work by Lekhnitskii [16] and Stroh [21], as reviewed
comprehensively by Ting [22]. Its relation to three-dimensional (3D)
anisotropic elasticity, where the fields are caused by point forces and
dislocation densities [14] has never been considered. Both generalized plane
strain and three-dimensional elasticity start with the appropriate 2D and 3D
differential equations and construct their solutions. In 3D, the fields vary
like the inverse distance between field and source points, in 2D the
dependence is logarithmic plus an angular term. Obviously, the 2D fields must
be obtainable from the 3D fields by integration along one direction, but
because of anisotropy evaluation of the necessary integral seems to be
impossible in real space. The objective of this text is to relate the 2D and
3D solutions (not the differential equations) by embedding the former into the
latter. This is done in reciprocal space using the so-called projection-slice
theorem (see, e.g., [8]).

Usually 3D and 2D problems are treated separately, the differential equation
for 2D is obtained by disallowing the partial derivative in one direction,
$\partial_{i}$ with $i=1,2,3$ becomes $\partial_{i}$ with $i=1,2$. In
electrostatics, for example, the 3D Green function
$G^{(3)}(\bm{R})=-(4\pi|\bm{R}|)^{-1}$ and the 2D Green function
$G^{(2)}(\bm{R})=(2\pi)^{-1}\ln|\bm{R}|$ are calculated independently of each
other. Physically, it is clear that integration of the 3D solutions over one
direction must produce the 2D solutions. Such 3D to 2D embedding is more
straightforward in Fourier space. Integration over ${\mathrm{d}}x_{3}$ and
resulting $x_{3}$-independence amounts to $k_{3}=0$ and concomitant
suppression of the ${\mathrm{d}}k_{3}$ integration. The simple example of
electrostatics serves to explain the idea. In potential theory, the Green
functions for the Laplace operator,

$\displaystyle\Delta G^{(3)}(\bm{R})=\delta^{(3)}(\bm{R})\,,\qquad\Delta
G^{(2)}(\bm{R})=\delta^{(2)}(\bm{R})\,,$ (1)

are (see [23])

$\displaystyle G^{(3)}(\bm{R})$ $\displaystyle=-\frac{1}{4\pi|\bm{R}|}$
$\displaystyle=-\frac{1}{(2\pi)^{3}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}}\,{\mathrm{e}}^{{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2}+k_{3}x_{3})}\,{\mathrm{d}}k_{1}{\mathrm{d}}k_{2}{\mathrm{d}}k_{3}\,,$
(2) $\displaystyle G^{(2)}(\bm{R})$
$\displaystyle=\frac{1}{2\pi}\,\ln|\bm{R}|$
$\displaystyle=-\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{k_{1}^{2}+k_{2}^{2}}\,{\mathrm{e}}^{{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}k_{1}{\mathrm{d}}k_{2}\,.$
(3)

In Fourier space, the integrands behave like $k^{-2}$ both in 3D and 2D, but
the different transforms without and with suppression of $k_{3}$ produce
$-(4\pi|\bm{R}|)^{-1}$ and $(2\pi)^{-1}\ln|\bm{R}|$. To descend from the 3D
solution $G^{(3)}(\bm{R})$ in real space, one constructs its Fourier
transform, uses the projection-slice theorem by putting $k_{3}=0$ in it and
transforms back to $G^{(2)}(\bm{R})$. The relation between 3D and 2D
differential equations is simple in real space, the relation between 3D and 2D
solutions is simple in Fourier space. In anisotropic elasticity, the
expressions are more complicated, the 2D and 3D solutions look distinctly
unrelated, their connection is far from obvious, but the same principle
prevails. The objective of the present paper is the application of the
projection-slice theorem to anisotropic elasticity.

##  2 Two-dimensional anisotropic elasticity

Here, we choose two orthogonal unit vectors $\bm{m}$ and $\bm{n}$ which are
normal to $\bm{t}$ such that $(\bm{m},\bm{n},\bm{t})$ forms a right-handed
Cartesian basis. This basis is rotated around $\bm{t}$ by an angle $\phi$
against another fixed $(\bm{m}_{0},\bm{n}_{0},\bm{t})$ basis, such that

$\displaystyle\begin{pmatrix}\bm{m}(\phi)\\\ \bm{n}(\phi)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos\phi&\sin\phi\\\
-\sin\phi&\cos\phi\end{pmatrix}\begin{pmatrix}\bm{m}_{0}\\\
\bm{n}_{0}\end{pmatrix}$ (4)

as shown in Fig. 1. Moreover, the field vector $\bm{x}$ reads with respect to
the unit basis $\bm{m}_{0}$ and $\bm{n}_{0}$

$\displaystyle\bm{x}$
$\displaystyle=r(\bm{m}_{0}\cos\omega+\bm{n}_{0}\sin\omega)\,,$ (5)

where $r=|\bm{x}|$ (see Fig 1).

$\bm{m}_{0}$$\bm{n}_{0}$$\bm{m}$$\bm{n}$$\bm{x}$$\ \phi$$\quad\omega$

Figure 1: The unit vectors $\bm{m}$ and $\bm{n}$ are to be turned
anticlockwise from $\bm{m}_{0}$ and $\bm{n}_{0}$ by an angle $\phi$, and the
vector $\bm{x}$ with angle $\omega$.

We consider a straight dislocation with Burgers vector $\bm{b}$ and a line
force with strength $\bm{F}$ located at the origin of the coordinate system.
The line defect runs along the axis $\bm{t}$. The fields of the straight
dislocation and line force are the displacement vector $\bm{u}$ and the stress
function vector $\bm{\Phi}$, the sources are the Burgers vector $\bm{b}$ and
the body force strength $\bm{F}$. Neither $\bm{u}$ nor $\bm{\Phi}$ are state
quantities. Nevertheless, $\bm{u}$ and $\bm{b}$ could be called the geometric
field and the geometric source, $\bm{\Phi}$ and $\bm{F}$ could be called the
physical field and the physical source. The geometrical and physical sources
contribute to both the geometric and physical fields. They are connected by
the following equation in integral form [12]

$\displaystyle\begin{pmatrix}\bm{u}(r,\omega)\\\
\bm{\Phi}(r,\omega)\end{pmatrix}$
$\displaystyle=-\frac{1}{(2\pi)^{2}}\left[\bm{I}\ln
r+\int_{0}^{\omega}\bm{N}(\phi)\,{\mathrm{d}}\phi\right]\int_{0}^{2\pi}\bm{N}(\phi)\,{\mathrm{d}}\phi\
\begin{pmatrix}\bm{b}\\\ -\bm{F}\end{pmatrix}\,,$ (6)

where $\bm{I}$ is the $6\times 6$ identity matrix. Moreover, Eq. (6) can be
written in rather compact form as (see [11, 22])

$\displaystyle\begin{pmatrix}\bm{u}(r,\omega)\\\
\bm{\Phi}(r,\omega)\end{pmatrix}$
$\displaystyle=-\left[\frac{1}{2\pi}\,\bm{I}\ln
r+\bm{\widetilde{N}}(\omega)\right]\bm{\widetilde{N}}(2\pi)\begin{pmatrix}\bm{b}\\\
-\bm{F}\end{pmatrix}\,$ (7)

with

$\displaystyle\bm{\widetilde{N}}(\omega)=\frac{1}{2\pi}\int_{0}^{\omega}\bm{N}(\phi)\,{\mathrm{d}}\phi\,.$
(8)

In this manner, Eq. (7) represents the six-dimensional unification of the
displacement fields of a straight dislocation and a line force, given by Asaro
et al. [3], and the stress functions of a straight dislocation and a line
force, given by Asaro et al. [4] in the framework of the integral formalism.
Eq. (7) is the field $[\bm{u}(r,\omega),\bm{\Phi}(r,\omega)]^{T}$ as function
of the sources $[\bm{b},-\bm{F}]^{T}$ at the origin. It was originally
obtained in terms of a specifically developed 2D “integral theory” without
reference to the 3D solution, unknown at the time (see [3, 4, 5, 12, 22]).

The matrix (8) fulfills the relation

$\displaystyle\bm{\widetilde{N}}(2\pi)\,\bm{\widetilde{N}}(2\pi)=-\bm{I}\,.$
(9)

Eq. (9) in conjunction with Eq. (7) insures the closure condition

$\displaystyle\begin{pmatrix}\bm{u}(r,2\pi)\\\
\bm{\Phi}(r,2\pi)\end{pmatrix}-\begin{pmatrix}\bm{u}(r,0\\\
\bm{\Phi}(r,0)\end{pmatrix}=\begin{pmatrix}\bm{b}\\\ -\bm{F}\end{pmatrix}\,.$
(10)

It is interesting to note that Eq. (6) is the solution of the equation (see
[2, 22])

$\displaystyle\left[\bm{I}\,\frac{1}{r}\,\partial_{\omega}-\bm{N}(\omega)\,\partial_{r}\right]\begin{pmatrix}\bm{u}(r,\omega)\\\
\bm{\Phi}(r,\omega)\end{pmatrix}=\begin{pmatrix}\bm{0}\\\
\bm{0}\end{pmatrix}\,.$ (11)

The $6\times 6$ matrix $\bm{N}(\phi)$ is defined by contraction of the elastic
constants with orthogonal unit vectors $\bm{m}$ and $\bm{n}$ according to
$(m\,m)_{ik}=m_{j}C_{ijkl}m_{l}$, $(m\,n)_{ik}=m_{j}C_{ijkl}n_{l}$, etc. As
mentioned above, the vectors $\bm{m}(\phi)$ and $\bm{n}(\phi)$ are turned
against the $\bm{m}_{0}$, $\bm{n}_{0}$ coordinate system by an angle $\phi$
(see Fig. 1), so that $\bm{N}$ depends on the angle $\phi$

$\displaystyle\bm{N}(\phi)=-\begin{pmatrix}(n\,n)^{-1}\,(n\,m)&(n\,n)^{-1}\\\
(m\,n)\,(n\,n)^{-1}\,(n\,m)-(m\,m)\ &(m\,n)\,(n\,n)^{-1}\end{pmatrix}\,.$ (12)

The structure of $\bm{N}(\phi)$ is such that it varies like a tangent of the
real angle $\phi$ [5], though its six complex eigenvalues vary like complex
tangents of a complex angle with six imaginary parts, the real part being
$\phi$,

$\displaystyle\frac{{\mathrm{d}}\bm{N}(\phi)}{{\mathrm{d}}\phi}+\bm{N}^{2}(\phi)+\bm{I}=\bm{0}\,.$
(13)

Although the solution (6) produces the correct closure condition (10), it is
not obvious how it does. To clarify the issue, a purely geometric field is
extracted by use of Lothe’s integral equation (Eq. (A.6) in Appendix)

$\displaystyle\begin{pmatrix}\bm{u}(r,\omega)\\\
\bm{\Phi}(r,\omega)\end{pmatrix}$
$\displaystyle=-\frac{1}{(2\pi)^{2}}\left[\int_{0}^{2\pi}\bm{N}(\phi)\,{\mathrm{d}}\phi\,\ln
r-2\pi\omega\,\bm{I}+\int_{0}^{2\pi}\bm{N}(\phi)\,\ln|\sin(\phi-\omega)|\,{\mathrm{d}}\phi\right]\begin{pmatrix}\bm{b}\\\
-\bm{F}\end{pmatrix}\,.$ (14)

The “const” in Eq. (A.6) giving a shift of $(\bm{u},\bm{\Phi})^{T}$ by a
multiple of $(\bm{b},-\bm{F})^{T}$ is irrelevant. Both in Eqs. (6) and (14)
there are discontinuities at $\omega=2\pi$, less obvious in (6) than in (14).
There is no discontinuity in the third term at any value of $\omega$, the
divergencies of the integrand are integrated over.

##  3 Three-dimensional solutions

For elastic homogeneity of an infinite anisotropic elastic medium the
expressions for $\bm{u}(\bm{\alpha})$, $\bm{u}(\bm{f})$ and
$\bm{\Phi}(\bm{\alpha})$ are given by Lazar and Kirchner [14]

$\displaystyle u_{i}(\bm{x})$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{\mathbb{R}^{3}}\frac{1}{|\bm{x}-\bm{y}|}\int_{0}^{2\pi}\epsilon_{mnr}C_{lkjn}\,\kappa_{m}\kappa_{k}(\kappa
C\kappa)_{il}^{-1}\,{\mathrm{d}}\phi\,\alpha_{jr}(\bm{y})\,{\mathrm{d}}\bm{y}\,,$
(15) $\displaystyle u_{i}(\bm{x})$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{\mathbb{R}^{3}}\frac{1}{|\bm{x}-\bm{y}|}\int_{0}^{2\pi}(\kappa
C\kappa)_{ij}^{-1}\,{\mathrm{d}}\phi\,f_{j}(\bm{y})\,{\mathrm{d}}\bm{y}\,,$
(16) $\displaystyle\Phi_{it}(\bm{x})$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{\mathbb{R}^{3}}\frac{1}{|\bm{x}-\bm{y}|}\int_{0}^{2\pi}\epsilon_{tkl}C_{ikmn}\epsilon_{npr}C_{qsjp}\kappa_{l}\kappa_{s}(\kappa
C\kappa)_{mq}^{-1}\,{\mathrm{d}}\phi\,\alpha_{jr}(\bm{y})\,{\mathrm{d}}\bm{y}$
$\displaystyle=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}\frac{1}{|\bm{x}-\bm{y}|}\,\big{(}C_{iljl}\delta_{tr}-C_{irjt}\big{)}\,\alpha_{jr}(\bm{y})\,{\mathrm{d}}\bm{y}$
$\displaystyle\quad-\frac{1}{8\pi^{2}}\int_{\mathbb{R}^{3}}\frac{1}{|\bm{x}-\bm{y}|}\int_{0}^{2\pi}\big{(}C_{ikml}C_{jkqs}\delta_{tr}-C_{irml}C_{jtqs}\big{)}\kappa_{l}\kappa_{s}(\kappa
C\kappa)_{mq}^{-1}\,{\mathrm{d}}\phi\,\alpha_{jr}(\bm{y})\,{\mathrm{d}}\bm{y}\,,$
(17)

where $\bm{\alpha}$ and $\bm{f}$ denote the dislocation density tensor and the
body force vector, respectively. $\bm{y}$ is the position of the sources.
Reciprocity is respected in (16) and (3),
$u_{i}(\bm{x})f_{i}(\bm{y})=u_{i}(\bm{y})f_{i}(\bm{x})$ and
$\alpha_{ij}(\bm{x})\Phi_{ij}(\bm{y})=\alpha_{ij}(\bm{y})\Phi_{ij}(\bm{x})$.
In reality, the one-dimensional integrations in Eqs. (15)–(3) are three-
dimensional Fourier transforms, reduced by the theorem [17] valid for
functions homogeneous in $k^{-2}$ and $|\bm{x}-\bm{y}|^{-1}$, which is here
the case. The unit vectors $\bm{\kappa}_{m}(\phi)$, normal to
$(\bm{x}-\bm{y})$, are turned around the direction of $(\bm{x}-\bm{y})$.

The expression (15) is not the complete displacement field caused by a
dislocation loop, a purely geometric contribution, $u_{i}=-b_{i}\Omega/4\pi$,
must be added (see [14]). $\Omega$ is the solid angle under which the
dislocation loop is seen. The set of equations (15)–(3) lacks the stress
function tensor caused by a body force. It is constructed from the fact [14]
that the Laplacian of the stress function $\bm{\Phi}$ is the curl of the
stress $\bm{\sigma}$,
$\Delta\bm{\Phi}=\text{curl}\,\bm{\sigma}=\text{curl}(\bm{C}:\text{grad}\,\bm{u})$.
With $\bm{u}(\bm{f})$ in Eq. (16) the result reads as

$\displaystyle\Phi_{it}(\bm{x})$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{\mathbb{R}^{3}}\frac{1}{|\bm{x}-\bm{y}|}\int_{0}^{2\pi}\epsilon_{tkl}C_{ikmn}\kappa_{l}\kappa_{n}(\kappa
C\kappa)_{mj}^{-1}\,{\mathrm{d}}\phi\,f_{j}({\bm{y}})\,{\mathrm{d}}\bm{y}\,.$
(18)

##  4 Projection-slice theorem: 3D $\rightarrow$ 2D projection

The projection-slice theorem (alternatively, the central-slice theorem or the
Fourier-slice theorem) is a powerful tool for the analysis of medical computed
tomography (CT) scans where a “projection” is an X-ray image of an internal
organ. It was originally derived in the context of radio astronomy in 1956
(see, e.g., [8]). The projection-slice theorem has also been used in acoustic
tomography [20] and imaging anisotropic media [9]. In this work, we will use,
for the first time, the projection-slice theorem in 3D anisotropic dislocation
theory.

######  Theorem.

In 3 dimensions, the projection-slice theorem states that the 2-dimensional
Fourier transform of the projection of a 3-dimensional function
$f(x_{1},x_{2},x_{3})$ onto a 2-dimensional linear submanifold is equal to a
2-dimensional slice of the 3-dimensional Fourier transform of that function
consisting of a 2-dimensional linear submanifold through the origin in the
Fourier space which is parallel to the projection submanifold. In operator
terms, if

  * •

${\cal F}_{2}$ and ${\cal F}_{3}$ are the 2- and 3-dimensional Fourier
transform operators,

  * •

${\cal P}_{2}$ is the projection operator (which projects a 3D function onto a
2D plane),

  * •

${\cal S}_{2}$ is a slice operator (which extracts a 2D central slice from a
3D function),

then it holds

$\displaystyle{\cal F}_{2}\,{\cal P}_{2}={\cal S}_{2}\,{\cal F}_{3}\,.$ (19)

Using the 2-dimensional inverse Fourier transform ${\cal F}^{-1}_{2}$ with
${\cal F}^{-1}_{2}\,{\cal F}_{2}=1$ on the projection-slice theorem (19), the
2-dimensional projection can be written

$\displaystyle{\cal P}_{2}={\cal F}^{-1}_{2}\,{\cal S}_{2}\,{\cal F}_{3}\,,$
(20)

which consists of the three steps: ${\cal F}_{3}$, ${\cal S}_{2}$ and ${\cal
F}^{-1}_{2}$.

The projection of the 3D function $f(x_{1},x_{2},x_{3})$ onto the $x_{1}x_{2}$
plane is given by the 2D function $f(x_{1},x_{2})$:

$\displaystyle f(x_{1},x_{2})$ $\displaystyle={\cal
P}_{2}\,f(x_{1},x_{2},x_{3})$
$\displaystyle=\int_{-\infty}^{\infty}f(x_{1},x_{2},x_{3})\,{\mathrm{d}}x_{3}\,,$
(21)

which is a 1D integral in the real space. The 3D Fourier transform of the 3D
function $f(x_{1},x_{2},x_{3})$ is given by

$\displaystyle\widehat{f}(k_{1},k_{2},k_{3})$ $\displaystyle={\cal
F}_{3}\,f(x_{1},x_{2},x_{3})$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x_{1},x_{2},x_{3})\,{\mathrm{e}}^{-{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2}+k_{3}x_{3})}\,{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}{\mathrm{d}}x_{3}\,.$
(22)

The 2D Fourier transform of the 2D function $f(x_{1},x_{2})$ reads as

$\displaystyle\widehat{f}(k_{1},k_{2})$ $\displaystyle={\cal
F}_{2}\,f(x_{1},x_{2})$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\
f(x_{1},x_{2})\,{\mathrm{e}}^{-{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}\,.$
(23)

The 2D slice of $\widehat{f}(k_{1},k_{2},k_{3})$ at $k_{3}=0$ is given by

$\displaystyle\widehat{f}(k_{1},k_{2},k_{3}=0)$ $\displaystyle={\cal
S}_{2}\,\widehat{f}(k_{1},k_{2},k_{3})$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x_{1},x_{2},x_{3})\,{\mathrm{e}}^{-{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}{\mathrm{d}}x_{3}$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}f(x_{1},x_{2},x_{3})\,{\mathrm{d}}x_{3}\right]{\mathrm{e}}^{-{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x_{1},x_{2})\,{\mathrm{e}}^{-{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}$
$\displaystyle={\cal F}_{2}\,f(x_{1},x_{2})\,,$ (24)

which is nothing but the 2D Fourier transform of the function
$f(x_{1},x_{2})$.

The inverse 2D Fourier transform of the 2D function $\widehat{f}(k_{1},k_{2})$
is defined by

$\displaystyle f(x_{1},x_{2})$ $\displaystyle={\cal
F}^{-1}_{2}\,\widehat{f}(k_{1},k_{2},)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\
\widehat{f}(k_{1},k_{2})\,{\mathrm{e}}^{{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}k_{1}{\mathrm{d}}k_{2}\,.$
(25)

Using Eq. (20) with Eqs. (4), (4) and (4), the projection (4) can be written
as

$\displaystyle f(x_{1},x_{2})$ $\displaystyle={\cal F}^{-1}_{2}\,{\cal
S}_{2}\,{\cal F}_{3}\,f(x_{1},x_{2},x_{3})$ $\displaystyle={\cal
F}^{-1}_{2}\,\widehat{f}(k_{1},k_{2},k_{3}=0)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\
\widehat{f}(k_{1},k_{2},k_{3}=0)\,{\mathrm{e}}^{{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}k_{1}{\mathrm{d}}k_{2}\,,$
(26)

which is a 2D integral in the Fourier space. Therefore, using Eq. (4), the 2D
projection of the 3D function $f(x_{1},x_{2},x_{3})$ in the $x_{3}$-direction
can be simply computed by means of the 2D inverse Fourier transform of the
slice $\widehat{f}(k_{1},k_{2},k_{3}=0)$ by substituting $k_{3}=0$ in the 3D
Fourier transform $\widehat{f}(k_{1},k_{2},k_{3})$.

##  5 Projection-slice theorem in 3D anisotropic elasticity

In 3D isotropic elasticity, the projection (4) can be easily computed in real
space. In fact, it was used in the calculation of the displacement fields of
straight screw and edge dislocations from the 3D fields in isotropic
elasticity by Mura [19]. However, in 3D anisotropic elasticity, the projection
using Eq. (4) is difficult and inconvenient. Therefore, in 3D anisotropic
elasticity, the projection using Eq. (4) is the proper tool for the projection
of 3D anisotropic elasticity toward 2D and will be used in this section.

The 2D fields can be obtained by using the projection given in Eq. (4)

$\displaystyle u_{i}(x_{1},x_{2})$ $\displaystyle={\cal
P}_{2}\big{[}u_{i}(x_{1},x_{2},x_{3})\big{]}\,,$ (27)
$\displaystyle\Phi_{it}(x_{1},x_{2})$ $\displaystyle={\cal
P}_{2}\big{[}\Phi_{it}(x_{1},x_{2},x_{3})\big{]}$ (28)

and by using the projection-slice theorem via Eq. (4)

$\displaystyle u_{i}(x_{1},x_{2})$ $\displaystyle={\cal F}^{-1}_{2}{\cal
S}_{2}\,{\cal F}_{3}\big{[}u_{i}(x_{1},x_{2},x_{3})\big{]}$
$\displaystyle={\cal
F}^{-1}_{2}\big{[}\widehat{u}_{i}(k_{1},k_{2},k_{3}=0)\big{]}\,,$ (29)
$\displaystyle\Phi_{it}(x_{1},x_{2})$ $\displaystyle={\cal F}^{-1}_{2}{\cal
S}_{2}\,{\cal F}_{3}\big{[}\Phi_{it}(x_{1},x_{2},x_{3})\big{]}$
$\displaystyle={\cal
F}^{-1}_{2}\big{[}\widehat{\Phi}_{it}(k_{1},k_{2},k_{3}=0)\big{]}\,.$ (30)

In the following, we will use Eqs. (5) and (5) for the projection of the 3D
fields (15)–(18) toward 2D fields.

###  5.1 First step: 3D Fourier transform

The 3D Fourier transforms of Eqs. (15)–(18) read as

$\displaystyle\widehat{u}_{i}(\bm{k})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{mnr}C_{lkjn}\,\kappa_{m}\kappa_{k}(\kappa
C\kappa)_{il}^{-1}\,\widehat{\alpha}_{jr}(\bm{k})\,,$ (31)
$\displaystyle\widehat{u}_{i}(\bm{k})$ $\displaystyle=\frac{1}{k^{2}}\,(\kappa
C\kappa)_{ij}^{-1}\,\widehat{f}_{j}(\bm{k})\,,$ (32)
$\displaystyle\widehat{\Phi}_{it}(\bm{k})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{tkl}C_{ikmn}\epsilon_{npr}C_{qsjp}\kappa_{l}\kappa_{s}(\kappa
C\kappa)_{mq}^{-1}\,\widehat{\alpha}_{jr}(\bm{k})\,,$ (33)
$\displaystyle\widehat{\Phi}_{it}(\bm{k})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{tkl}C_{ikmn}\kappa_{l}\kappa_{n}(\kappa
C\kappa)_{mj}^{-1}\,\widehat{f}_{j}({\bm{k}})\,,$ (34)

where $\bm{k}\in\mathbb{R}^{3}$ and $k=\sqrt{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}}$.

###  5.2 Second step: 2D slice at $k_{3}=0$

In Fourier space, the slices of Eqs. (31)–(34) at $k_{3}=0$ become

$\displaystyle\widehat{u}_{i}(k_{1},k_{2})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{mnr}C_{lkjn}\,\kappa_{m}\kappa_{k}(\kappa
C\kappa)_{il}^{-1}\,\widehat{\alpha}_{jr}(k_{1},k_{2})\,,$ (35)
$\displaystyle\widehat{u}_{i}(k_{1},k_{2})$
$\displaystyle=\frac{1}{k^{2}}\,(\kappa
C\kappa)_{ij}^{-1}\,\widehat{f}_{j}(k_{1},k_{2})\,,$ (36)
$\displaystyle\widehat{\Phi}_{it}(k_{1},k_{2})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{tkl}C_{ikmn}\epsilon_{npr}C_{qsjp}\kappa_{l}\kappa_{s}(\kappa
C\kappa)_{mq}^{-1}\,\widehat{\alpha}_{jr}(k_{1},k_{2})\,,$ (37)
$\displaystyle\widehat{\Phi}_{it}(k_{1},k_{2})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{tkl}C_{ikmn}\kappa_{l}\kappa_{n}(\kappa
C\kappa)_{mj}^{-1}\,\widehat{f}_{j}(k_{1},k_{2})\,,$ (38)

where $\bm{k}\in\mathbb{R}^{2}$, $k=\sqrt{k_{1}^{2}+k_{2}^{2}}$ and
$\bm{\kappa}=\bm{k}/k\in\mathbb{R}^{2}$.

Using the fact that $\bm{\kappa}\in\mathbb{R}^{2}$ and the major symmetry of
the elastic tensor $C_{ijkl}=C_{klij}$, Eqs. (35), (37) and (38) can be
written

$\displaystyle\widehat{u}_{i}(k_{1},k_{2})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{3\alpha\beta}C_{j\alpha
l\gamma}\,\kappa_{\beta}\kappa_{\gamma}(\kappa
C\kappa)_{il}^{-1}\,\widehat{\alpha}_{j3}(k_{1},k_{2})\,,$ (39)
$\displaystyle\widehat{\Phi}_{i3}(k_{1},k_{2})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{3\beta\alpha}C_{i\alpha
l\gamma}\epsilon_{3\gamma\nu}C_{r\mu j\nu}\kappa_{\beta}\kappa_{\mu}(\kappa
C\kappa)_{lr}^{-1}\,\widehat{\alpha}_{j3}(k_{1},k_{2})\,,$ (40)
$\displaystyle\widehat{\Phi}_{i3}(k_{1},k_{2})$
$\displaystyle=-\frac{1}{k^{2}}\,\epsilon_{3\beta\alpha}C_{i\alpha
l\gamma}\kappa_{\beta}\kappa_{\gamma}(\kappa
C\kappa)_{lj}^{-1}\,\widehat{f}_{j}(k_{1},k_{2})\,,$ (41)

where the Latin indices take the values of 1, 2 and 3, whereas the Greek
indices take the values of 1 and 2 only. In 2D, the indices $t$ and $r$ became
3. There remain 3 displacement components $u_{i}=u_{i}(x_{1},x_{2})$ and 3
stress functions $\Phi_{i3}=\Phi_{i3}(x_{1},x_{2})$.

Now, Eqs. (39), (36), (40) and (41) can be written in the following matrix
equation form

$\displaystyle\begin{pmatrix}\widehat{u}_{i}(k_{1},k_{2})\\\
\widehat{\Phi}_{i3}(k_{1},k_{2})\end{pmatrix}$
$\displaystyle=-\frac{1}{k^{2}}\begin{pmatrix}\epsilon_{3\alpha\beta}\kappa_{\beta}C_{j\alpha
l\gamma}\kappa_{\gamma}(\kappa\,\kappa)^{-1}_{il}&(\kappa\,\kappa)^{-1}_{ij}\\\
\epsilon_{3\beta\alpha}C_{i\alpha l\gamma}\epsilon_{3\gamma\nu}C_{r\mu
j\nu}\kappa_{\beta}\kappa_{\mu}(\kappa\,\kappa)^{-1}_{lr}&\epsilon_{3\alpha\beta}\kappa_{\beta}C_{i\alpha
l\gamma}\kappa_{\gamma}(\kappa\,\kappa)^{-1}_{lj}\end{pmatrix}\begin{pmatrix}\widehat{\alpha}_{j3}(k_{1},k_{2})\\\
-\widehat{f}_{j}(k_{1},k_{2})\end{pmatrix}\,.$ (42)

Identifying $\bm{\kappa}=\bm{n}$ and
$\epsilon_{3\alpha\beta}\kappa_{\beta}=\epsilon_{3\alpha\beta}n_{\beta}=m_{\alpha}$
(see Fig. 1), the north-west (NW) block of the matrix in Eq. (42) becomes

$\displaystyle\epsilon_{3\alpha\beta}\kappa_{\beta}C_{j\alpha
l\gamma}\kappa_{\gamma}(\kappa\,\kappa)^{-1}_{il}=m_{\alpha}C_{j\alpha
l\gamma}n_{\gamma}(n\,n)^{-1}_{il}=(m\,n)_{jl}(n\,n)^{-1}_{il}=(n\,n)^{-1}_{il}(n\,m)_{lj}\,,$
(43)

which is the NW block of $\bm{N}$ given in Eq. (12). The north-east (NE) block
of the matrix in Eq. (42) is simply
$(\kappa\,\kappa)^{-1}_{ij}=(n\,n)^{-1}_{ij}$, being the NE block in Eq. (12).
The south-east (SE) block of the matrix in Eq. (42) becomes

$\displaystyle\epsilon_{3\alpha\beta}\kappa_{\beta}C_{i\alpha
l\gamma}\kappa_{\gamma}(\kappa\,\kappa)^{-1}_{lj}=m_{\alpha}C_{i\alpha
l\gamma}n_{\gamma}(n\,n)^{-1}_{lj}=(m\,n)_{il}(n\,n)^{-1}_{lj}\,,$ (44)

which is the SE block of $\bm{N}$ given in Eq. (12). The south-west (SW) block
of the matrix in Eq. (42) becomes

$\displaystyle\epsilon_{3\beta\alpha}C_{i\alpha
l\gamma}\epsilon_{3\gamma\delta}C_{r\mu
j\delta}\,\kappa_{\beta}\kappa_{\mu}(\kappa\,\kappa)^{-1}_{lr}=C_{i\alpha
l\beta}n_{\beta}C_{j\alpha r\mu}n_{\mu}(n\,n)^{-1}_{lr}-C_{i\alpha j\alpha}$
$\displaystyle\hskip
85.35826pt=(m\,n)_{il}(m\,n)_{jr}(n\,n)^{-1}_{lr}+(n\,n)_{il}(n\,n)_{jr}(n\,n)^{-1}_{lr}-(m\,m)_{ij}-(n\,n)_{ij}$
$\displaystyle\hskip
85.35826pt=(m\,n)_{il}(n\,n)^{-1}_{lr}(n\,m)_{rj}-(m\,m)_{ij}\,,$ (45)

being the SW block of $\bm{N}$ given in Eq. (12). In this way, the matrix
equation (42) can be written in terms of the $6\times 6$ matrix $\bm{N}(\phi)$
given in Eq. (12)

$\displaystyle\begin{pmatrix}\widehat{\bm{u}}(k_{1},k_{2})\\\
\widehat{\bm{\Phi}}(k_{1},k_{2})\end{pmatrix}$
$\displaystyle=\frac{1}{k^{2}}\,\bm{N}(\phi)\begin{pmatrix}\widehat{\bm{\alpha}}(k_{1},k_{2})\\\
-\widehat{\bm{f}}(k_{1},k_{2})\end{pmatrix}\,.$ (46)

In the Fourier space, the dislocation density and the line force density of
straight line defects are given by $\widehat{\bm{\alpha}}=\bm{b}$ and
$\widehat{\bm{f}}=\bm{F}$, which are the Fourier transforms of
$\bm{\alpha}(x_{1},x_{2})=\bm{b}\,\delta(x_{1})\delta(x_{2})$ and
$\bm{f}(x_{1},x_{2})=\bm{F}\,\delta(x_{1})\delta(x_{2})$, and the matrix
equation (46) becomes for straight line defects

$\displaystyle\begin{pmatrix}\widehat{\bm{u}}(k_{1},k_{2})\\\
\widehat{\bm{\Phi}}(k_{1},k_{2})\end{pmatrix}$
$\displaystyle=\frac{1}{k^{2}}\,\bm{N}(\phi)\begin{pmatrix}\widehat{\bm{b}}\\\
-\widehat{\bm{F}}\end{pmatrix}\,.$ (47)

This equation clarifies the meaning of the $6\times 6$ matrix $\bm{N}(\phi)$,
which is the central quantity of the integral theory: In reciprocal space it
is the coefficient between fields and sources, only the 2D inverse Fourier
transform produces the characteristic logarithmic plus angular dependence.

###  5.3 Third step: 2D inverse Fourier transform

The 2D inverse Fourier transform of Eq. (47) reads in Cartesian coordinates

$\displaystyle\begin{pmatrix}\bm{u}(x_{1},x_{2})\\\
\bm{\Phi}_{3}(x_{1},x_{2})\end{pmatrix}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\,\frac{1}{k^{2}}\,\bm{N}(\phi)\,{\mathrm{e}}^{{\mathrm{i}}(k_{1}x_{1}+k_{2}x_{2})}\,{\mathrm{d}}k_{1}{\mathrm{d}}k_{2}\,\begin{pmatrix}\bm{b}\\\
-\bm{F}\end{pmatrix}\,$ (48)

and in polar coordinates it becomes

$\displaystyle\begin{pmatrix}\bm{u}(x_{1},x_{2})\\\
\bm{\Phi}(x_{1},x_{2})\end{pmatrix}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\bm{N}(\phi)\int_{0}^{\infty}\,\frac{1}{k}\,{\mathrm{e}}^{{\mathrm{i}}k\,\bm{n}\cdot\bm{x}}\,{\mathrm{d}}k\,{\mathrm{d}}\phi\,\begin{pmatrix}\bm{b}\\\
-\bm{F}\end{pmatrix}\,.$ (49)

The $k$-integration can be carried out by using the principal value integral
(see Eq. (32) in Sec. 9 in [23], and Eq. (33c) in Sec. 6.4 in [10])

$\displaystyle\frac{1}{2}\,{\cal{P}}\int_{-\infty}^{\infty}\frac{1}{k}\,{\mathrm{e}}^{{\mathrm{i}}kt}\,\text{d}k={\cal{P}}\int_{0}^{\infty}\frac{1}{k}\,{\mathrm{e}}^{{\mathrm{i}}kt}\,\text{d}k=-\gamma-\ln|t|\,,$
(50)

where $\gamma$ denotes the Euler constant and ${\cal{P}}$ means the principal
value. Another expression for the integral in Eq. (50) is the limit
$z\rightarrow 0$ of the exponential integral $E_{1}(z)$, formula 5.1.1 of
Abramowitz and Segun [1]. The constant term $\gamma$ can be neglected because
it gives just an irrelevant constant displacement and stress function. Using
Eq. (50), Eq. (49) reduces to

$\displaystyle\begin{pmatrix}\bm{u}(x_{1},x_{2})\\\
\bm{\Phi}(x_{1},x_{2})\end{pmatrix}$
$\displaystyle=-\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\bm{N}(\phi)\,\ln|\bm{x}\cdot\bm{n}|\,{\mathrm{d}}\phi\,\begin{pmatrix}\bm{b}\\\
-\bm{F}\end{pmatrix}\,.$ (51)

For the geometry given in Fig. 1 and using Eqs. (4) and (5), the inner product
between the vectors $\bm{x}$ and $\bm{n}$ reads as

$\displaystyle\bm{x}\cdot\bm{n}=r\sin(\omega-\phi)=-r\sin(\phi-\omega)\,,$
(52)

so that the $\ln$-term in Eq. (51) can be written as

$\displaystyle\ln|\bm{x}\cdot\bm{n}|=\ln r+\ln|\sin(\phi-\omega)|\,.$ (53)

Substituting Eq. (53) into Eq. (51), we obtain in polar coordinates

$\displaystyle\begin{pmatrix}\bm{u}(r,\omega)\\\
\bm{\Phi}(r,\omega)\end{pmatrix}$
$\displaystyle=-\frac{1}{(2\pi)^{2}}\left[\int_{0}^{2\pi}\bm{N}(\phi)\,{\mathrm{d}}\phi\,\ln
r+\int_{0}^{2\pi}\bm{N}(\phi)\,\ln|\sin(\phi-\omega)|\,{\mathrm{d}}\phi\right]\begin{pmatrix}\bm{b}\\\
-\bm{F}\end{pmatrix}\,.$ (54)

This is the non-$\omega$ part of the generalized plane strain solution (14),
it completes the embedding. The geometric $\omega$ part in Eq. (14) is due to
the defect-topology of the line defect and its 3D version. For instance, the
purely geometric term $\Omega$ of a dislocation loop in 3D becomes $-2\omega$
of a straight dislocation in 2D (see Appendix, Eq. (B)).

##  6 Conclusion

Hitherto the field equations of generalized plane strain and the 3D field
equations were solved separately, the relation between their solution remained
uninvestigated. The question, how the fields of generalized plane strain are
imbedded into the 3D fields, remained open. The answer, if such a reduction of
the 3D to the 2D solution is possible, is given in the affirmative.
Generalized plane strain is the projection of the 3D solution into 2D. This
projection, impossible in real space, is achieved by slicing in reciprocal
space by means of the projection-slice theorem [8]. A relative of the Fourier-
slice theorem, the Radon transform, was used by Bacon et al. [5] for
calculating derivatives of the Green function. However, the Radon transform,
$\cal R$, is the integral transform which takes a function $f$ defined on a
plane to a function ${\cal{R}}[f]$ defined on the 2D space of lines in the
plane, whose value at a particular line is equal to the line integral over the
function over that line. The inversion of ${\cal{R}}[f]$ to $f$ is possible.

Extension of the presented sextic ($6\times 6$) formalism of anisotropic
elasticity to the decadic ($10\times 10$) formalism of piezoelectric
piezomagnetic elastic media [2] and piezoelectric-piezomagnetic-
magnetoelectric elastic media [13] is possible and straightforward.

The projection takes the 3D situation with point forces and dislocation
densities to the corresponding 2D one, in both the medium is of infinite
extent, no boundary conditions are involved. Other 2D situations, like plane
strain or plane stress have no 3D equivalent. Because they are already
constrained by geometric conditions, like invariance of components or
vanishing derivatives in one direction, they are by definition specifically 2D
geometries. Finite 2D problems with boundary conditions have no nontrivial 3D
equivalent either.

In medical imaging, the projection of the 3D situation is a 2D X-ray picture
taken. In medicine, the inverse tomographic reconstruction of the 3D situation
from 2D projections taken in many different direction is of concern. The
elastic analogue of reconstructing 3D solutions from 2D integral solutions on
many different planes remains an interesting, albeit artificial problem.

The slicing of the Fourier-slice theorem elucidates several features of the 2D
vs 3D situation. For example, the vanishing trace of the 3D stress function,
trace$(\bm{\Phi})=0$ in 3D, corresponds to zero divergence,
$\text{div}\,\bm{\Phi}=0$, of the stress vector in 2D. The embedding clarifies
the nature of the so-called six-dimensional integral theory, it is a 2D
inverse Fourier transform obtained as slice of a 3D Fourier transform. The
“integral formalism” is more natural than the preceding Stroh and Lekhnitzki
theories formulated in terms of eigenvectors and eigenvalues, these are
spectral representations of the “integral theory” in Fourier space.

## Acknowledgement

Markus Lazar gratefully acknowledges the grant from the Deutsche
Forschungsgemeinschaft (Grant number LA1974/4-2).

##  Appendix A Lothe’s integral equation

The eigenvalues of $\bm{N}(\omega)$ behave like tangents of a complex angle,
$p_{\alpha}(\omega)=\tan(\Psi_{\alpha}-\omega)$ (see Eq. 3.5.34 in [5]) with
negative sign for the real angle. Apply the sum formula for the tangent of the
complex argument ($\Psi_{\alpha}-\omega$) to two of these. The integrals are
taken in the principal value sense

$\displaystyle[1+p_{\alpha}(\phi)\,p_{\alpha}(\omega)]=-[p_{\alpha}(\phi)-p_{\alpha}(\omega)]\cot(\phi-\omega)\,,$
(A.1)

where $\alpha=1,\dots,6$. Integrate over ${\mathrm{d}}\phi$ from 0 to $2\pi$

$\displaystyle
2\pi+\int_{0}^{2\pi}p_{\alpha}(\phi)\,{\mathrm{d}}\phi\,p_{\alpha}(\omega)$
$\displaystyle=-\int_{0}^{2\pi}p_{\alpha}(\phi)\cot(\phi-\omega)\,{\mathrm{d}}\phi+\int_{0}^{2\pi}p_{\alpha}(\omega)\cot(\phi-\omega)\,{\mathrm{d}}\phi$
$\displaystyle=-\int_{0}^{2\pi}p_{\alpha}(\phi)\cot(\phi-\omega)\,{\mathrm{d}}\phi\,.$
(A.2)

The arguments of $p_{\alpha}(\omega)$ and $p_{\alpha}(\phi)$ have the same
complex part, therefore the cotangent in (A) has a purely real argument. The
second integral is zero. One more integration over ${\mathrm{d}}\omega$ from 0
to $2\pi$ gives a central result of the integral theory

$\displaystyle\left[\int_{0}^{2\pi}p_{\alpha}(\phi)\,{\mathrm{d}}\phi\right]^{2}=(2\pi)^{2}\,.$
(A.3)

Multiplication with the $\omega$-independent eigenvectors of $\bm{N}$ confirms
Eq. (9) above,

$\displaystyle\left[\int_{0}^{2\pi}\bm{N}(\phi)\,{\mathrm{d}}\phi\right]^{2}=(2\pi)^{2}\bm{I}\,.$
(A.4)

Multiplied with the $\omega$-independent eigenvectors of $\bm{N}$ to obtain
the spectral form, the integral equation (A) for the tangent of a complex
argument becomes an integral equation for a matrix that varies like a tangent
(see Eq. (114) in Lothe [18])

$\displaystyle-\frac{1}{2\pi}\,{\cal{P}}\int_{0}^{2\pi}\bm{N}(\phi)\cot(\phi-\omega)\,{\mathrm{d}}\phi=\bm{I}+\bm{N}(\omega)\,\int_{0}^{2\pi}\bm{N}(\phi)\,{\mathrm{d}}\phi\,.$
(A.5)

Integrated over ${\mathrm{d}}\omega$

$\displaystyle\int_{0}^{2\pi}\bm{N}(\phi)\ln|\sin(\phi-\omega)|\,{\mathrm{d}}\phi=2\pi\omega\bm{I}+\left(\int_{0}^{\omega}\bm{N}(\phi)\,{\mathrm{d}}\phi\right)\int_{0}^{2\pi}\bm{N}(\phi)\,{\mathrm{d}}\phi+\text{const}\,.$
(A.6)

For no value of $\omega$ a discontinuity is present in Eq. (A.6). The
discontinuities at $\omega=2\pi$ of both the first and second term on the
right-hand side cancel, the term on the left-hand side is just an integral for
any value of $\omega$ has no discontinuity for any $\omega$.

##  Appendix B Projection of the 3D solid angle $\Omega$

The 3D solid angle reads for a dislocation loop (see [14, 15, 18])

$\displaystyle\Omega(x_{1},x_{2},x_{3})=-\int_{S}\partial_{m}\frac{1}{R}\,{\mathrm{d}}S^{\prime}_{m}\,,$
(B.1)

subtended by the loop area $S^{\prime}_{m}$.

For a straight dislocation with dislocation line direction along the
$x_{3}$-axis, $S$ can be chosen as a semi-infinite plane
$x^{\prime}_{1}x^{\prime}_{3}$ for $x^{\prime}_{1}>0$ at $x^{\prime}_{2}=0$
with
${\mathrm{d}}S^{\prime}_{2}={\mathrm{d}}x^{\prime}_{1}\,{\mathrm{d}}x^{\prime}_{3}$
and $m=2$, the solid angle becomes

$\displaystyle\Omega(x_{1},x_{2})=\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{x_{2}}{R^{3}}\,{\mathrm{d}}x^{\prime}_{3}\,{\mathrm{d}}x^{\prime}_{1}\,,$
(B.2)

where $R=[(x_{1}-x^{\prime}_{1})^{2}+x_{2}^{2}+(x^{\prime}_{3})^{2}]^{1/2}$.
After integration over ${\mathrm{d}}x^{\prime}_{3}$

$\displaystyle\int_{-\infty}^{\infty}\frac{1}{R^{3}}\,{\mathrm{d}}x^{\prime}_{3}=\frac{2}{(x_{1}-x^{\prime}_{1})^{2}+x_{2}^{2}}\,,$
(B.3)

and further integration over ${\mathrm{d}}x^{\prime}_{1}$ the solid angle
(B.2) reduces to (see also [15, 6])

$\displaystyle\Omega(x_{1},x_{2})$
$\displaystyle=2\int_{0}^{\infty}\frac{x_{2}}{(x_{1}-x^{\prime}_{1})^{2}+x_{2}^{2}}\,{\mathrm{d}}x^{\prime}_{1}$
$\displaystyle=-2\arctan\left(\frac{x_{2}}{x_{1}}\right)$
$\displaystyle=-2\,\omega(x_{1},x_{2})\,,$ (B.4)

which is nothing but twice the negative polar angle $\omega$. The $4\pi$
discontinuity of $\Omega$ becomes a $2\pi$ discontinuity of $\omega$. The
purely geometric part of the displacement field of a straight dislocation is

$\displaystyle\bm{u}=-\frac{\bm{b}\,\Omega}{4\pi}=\frac{\bm{b}\,\omega}{2\pi}\,$
(B.5)

as in Eq. (14).

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Support Vector Regression Parameters Optimization using Golden Sine Algorithm and its application in stock market
Mohammadreza Ghanbari, Mahdi Goldani

mode = title]Support Vector Regression Parameters Optimization using Golden Sine Algorithm and its application in stock market

1]Mohammadreza Ghanbari

[1]Department of Mathematical Sciences , Sharif University of Technology, Tehran, Iran

2]Mahdi Goldani
[1]

[2]Department of Economics, Hakim Sabzevari University, Sabzevar, Iran

Support vector machine modeling is a new approach in machine learning for classification showing good performance on forecasting problems of small samples and high dimensions. Later, it promoted to Support Vector Regression (SVR) for regression problems. A big challenge for achieving reliable is the choise of appropriate parameters. Here, a novel Golden sine algorithm (GSA) based SVR is proposed for proper selection of the parameters. For comparison, the performance of the proposed algorithm is compared with eleven other meta-heuristic algorithms on some historical stock prices of technological companies from Yahoo Finance website based on Mean Squared Error and Mean Absolute Percent Error. The results demonstrate that the given algorithm is efficient for tuning the parameters and is indeed competitive in terms of accuracy and computing time.

Support Vector Regression Meta-heuristics Optimization Algorithms Golden Sine Algorithm Parameter Tuning Time Series Prediction

§ INTRODUCTION

Support Vector Machine (SVM) was originally introduced by Vapnik [Cortes and Vapnik, 1995] for classification problems. It can potentially solve small-sample, non-linear and high dimensions problems by using structural risk minimization principle instead of the empirical risk principle. Essentially, the SVM is a convex quadratic programming method by which it is possible to find the global,  rather than local optimum. After on, SVM was developed and extended to Support Vector Regression (SVR) to solve regression problems [Cortes and Vapnik, 1995]. SVR is arguably one of the best techniques and experimental results show considerable performance compared to other nonlinear methods [Thissen et al., 2003, Vapnik, 2013]. However, setting of the parameters for the SVR plays a significant role and the performance accuracy changes considerably upon a bad choice of parameters [Chapelle et al., 2002, Duan et al., 2003, Kwok, 2000, Yeh et al., 2011].

Concerning this issue, a main approach to select the SVR parameters optimally is to make use of an optimization technique for finding  optimal values. Generally, three common techniques to optimize the SVR parameters are grid search [Hsu et al., 2003], which in practical applications, it is usually vulnerable to get to a local optimum, gradient descent [Keerthi et al., 2007] and meta-heuristics algorithms [Blum and Roli, 2003, Talbi, 2009].

Meta-heuristics algorithms have been introduced that may provide a sufficiently good solution to an optimization problem specially on presence of incomplete information or limited computing capacity. They have shown superior results in the case of solving optimization problems for parameter tuning of complex models   [Gogna and Tayal, 2013].
In the past, many meta-heuristics
algorithms have been proposed for the selection of optimal SVR parameters. Example are, Genetic Algorithm (GA) [Huang, 2012, Gu et al., 2011, Min et al., 2006, Wu et al., 2009], Grey Wolf Optimizer (GWO) [Mustaffa et al., 2015], Particle Swarm Optimization (PSO) [Li-Xia et al., 2011, Wu, 2010],
Sine Cosine Algorithm (SCA) [Li et al., 2018],
Butterfly Optimization Algorithm (BOA) [Ghanbari and Arian, 2019],
Firefly Algorithm (FA) [Kavousi-Fard et al., 2014] and
Bat Algorithm (BA) [Tavakkoli et al., 2015].

Recently, a novel math-based meta-heuristic optimization algorithm inspired by sine function, named as Golden Sine Algorithm (GSA), was designed by [Tanyildizi and Demir, 2017]. The GSA algorithm searches to approach a better solution in each iteration by trying to bring the current point closer to the target value and the solution space gets to be narrowed by the golden section algorithm so that the areas with supposedly good results instead of the whole solution space are examined.

Here, we propose a novel GSA based SVR model where GSA is used to set the parameters of SVR. For validation, eleven other meta-heuristic algorithms, namely Whale Optimization Algorithm (WOA), Salp Swarm Algorithm (SSA), Neural Network Algorithm (NNA), Firefly Algorithm (FA), Multi-Verse Optimizer (MVO), Moth-Flame Optimization (MFO), Harris Hawks Optimization (HHO), Grey Wolf Optimization (GWO), Butterfly Optimization Algorithm (BOA), Biogeography-Based Optimization (BBO) and Artificial Bee Colony Optimization (ABC) are appropriated to SVR to optimize the parameters, are compared with the proposed algorithm.

The remainder of our work is as follows: In Section <ref>, we discuss the presented GSA-SVR model, the support vector regression and GSA. In Section <ref>, the  model is tested on some datasets and compared
with other models and the obtained experimental results are discussed. Conclusions and future research directions are provided in Section <ref>.

§ MATERIALS AND METHODS

In this section, we briefly discuss about Support Vector Regression (SVR) and the Golden Sine Algorithm (GSA).

§.§ Support Vector Regression

Support vector machine (SVM) is a machine learning algorithm introduced by Vapnik in 1995 for classification problems. It has been one of the more widely used methods in recent years as a powerful method. It was first used to address a binary pattern classification problem. Then, it was promoted to support vector regression (SVR) for regression problems by using $\epsilon$-insensitive loss function to penalize data when they are greater than $\epsilon$ [Cortes and Vapnik, 1995]. SVR aims to provide a nonlinear mapping function to map the training dataset to a high dimensional feature space [Yeh et al., 2011].

The given training dataset is
$\{(x_i,y_i)\}_{i = 1}^n$, where
$x_i \in \mathbb{R}^d$
is input data,
$y_i \in \mathbb{R}$ is the output value of the $i$-th data point in the dataset, $d$ is the dimension of samples and $n$ is the number of samples. The nonlinear function between
the input and the output is formulated as:

\begin{align} \label{2.1}
y=f(x)=w^{T} \phi(x)+b,
\end{align}

where $\phi: \mathbb{R}^d \longrightarrow F$ is a nonlinear mapping to the feature space, $w \in F$ is a vector of weight coefficients and $b$ is a bias constant. The $w$ and $b$ are estimated by minimizing the following optimization problem:

\begin{equation}
\begin{split}
\text{Min} & \frac{1}{2} ||w||^2, \\
\text{s.t. } &
\begin{cases}
y_i-w^T\phi(x_i)-b  & \leq \epsilon,\\
y_i-w^T\phi(x_i)-b  & \geq -\epsilon,
\end{cases}
\end{split}
\end{equation}

The slack variables $\xi$ and $\xi^*$ are used to penalize points from $\epsilon$-insensitive band:

\begin{equation}
\begin{split}\label{2.3}
\text{Min} & \frac{1}{2} ||w||^2+ C\sum_{i=1}^n(\xi_i+\xi_i^*),\\
\text{s.t. } &
\begin{cases}
y_i-w^T\phi(x_i)-b  \leq \epsilon + \xi_i,\\
y_i-w^T\phi(x_i)-b  \geq -\epsilon - \xi_i^*,\\
\xi_i,\xi_i^* \geq 0, i = 1,\cdots, n,
\end{cases}
\end{split}
\end{equation}

where $C$ is a constant known as the penalty parameter to specify the
trade-off between the empirical risk and regularization terms, $\epsilon$ is the insensitive loss function and the slack variables $\xi_i$ and $\xi_i^*$, correspond to upper and lower deviations, respectively, and $n$ is the number of training

Using the Lagrangian and corresponding optimality conditions, the obtained generic equation is written as [Smola and Schölkopf, 2004, Cortes and Vapnik, 1995]:

\begin{align}
\end{align}

where $\beta_i$ and $\beta_i^*$ are nonzero Lagrange multipliers and
$K(x_i,x)$ is the kernel function. In our work, Radial Basis Function (RBF) has been used as kernel function:

\begin{align}
K(x_i,x_j) = \exp(-\gamma||x_i-x_j||^2),
\end{align}

where $\gamma$ is the RBF width parameter.

§.§ Golden Sine Algorithm

Golden Sine Algorithm (GSA) is a novel math-based meta-heuristic optimization algorithm inspired by sine function for solving optimization problems [Tanyildizi and Demir, 2017].
Sine, a trigonometric function, is the coordinate relative to the $y$-axis of a point on a $1$-unit radius circle that is the central origin. An orthogonal triangle with an angle made by the $y$-axis of a straight line drawn from the origin or with the same angle is calculated with the hypotenuse section of the edge opposite this angle. The defining range of the function is $[-1, 1]$.
The scan of the unit circle of all values of the sine
function is similar to the search of the search space in
optimization problems. This similarity has inspired the
development of GSA. The operator used in the algorithm is
shown by

\begin{equation}\label{eq: 2.3.1}
V_{ij}=V_{ij}\left|\sin \left(r_{1}\right)\right|-r_{2} \sin \left(r_{1}\right)\left|x_{1} D_j-x_{2} V_{ij}\right|
\end{equation}

where $V_{ij}$ is the value of current solution in the $i$-th dimension, $D$ is the determined target value, $r_1$ is a random number in the range $[0, 2\pi]$ and $r_2$ is a random number in the range $[0, \pi]$, and $x_1$ and $x_2$ are the coefficients obtained by the golden section method. These coefficients limit the search space and also allow the current value to approach the target value.

The wide range of the search space is a major problem
for solving problems. The effect of limiting the search
space in solving problems is significantly affecting the
results. GSA uses the golden section method to make
this process the best possible way. Golden section search is an optimization technique that can be used to find the maximum or minimum value of a single unimodal function. The name is from the golden ratio. Two numbers, $p$ and $q$, are in a golden ratio if

\begin{equation}
\frac{p+q}{p}=\frac{p}{q}=\tau,
\end{equation}

or equivalently,

\begin{equation}
\end{equation}

\begin{equation}\label{eq: 2.3.5}
\end{equation}

Solving Eq. <ref>, we get the positive root as

\begin{equation}\label{eq: 2.3.6}
\tau=\frac{1-\sqrt{5}}{2} \approx 0.618033,
\end{equation}

here $\tau$ is called the golden number.

In GSA, initial default values for $a$ and $b$ are
considered to be $-\pi$ and $\pi$, respectively. These two
coefficients are applied to the current and target values in
the first iteration. Then, the coefficients $x_1$ and $x_2$ are
updated as the target value changes.

To avoid the situation of equality for $x_1$ and $x_2$, an equality check is performed. It means that if the two values are equal, then the random numbers $rand_1$ and  $rand_2$ are generated in the range, respectively, $[0, \pi]$ and $[0, -\pi]$ and $x_1$ and $x_2$ are recalculated.

§.§ GSA for Parameter Optimization of SVR

Before presentation the algorithm, we first discuss the data pre-processing of time series.

Phase Space Reconstruction [Takens, 1981] is a method in which uncover the hidden information embedded in the time series dynamics. This method provides a simplified multidimensional representation of data.

Let $ \{x_i\}_{i=1}^n$ represent an $n$ point time series.
Then, the reconstructed phase space can be expressed as a matrix as follows:

\begin{equation}
X =
\begin{bmatrix}\label{xtakens}
x_1 & x_{1+\tau} & \cdots &  x_{1+(m-1)\tau}\\
x_2 & x_{2+\tau} & \cdots & x_{2+(m-1)\tau}\\
\vdots & \vdots & \ddots & \vdots \\
x_{n-1-(m-1)\tau} & x_{n-1-(m-2)\tau} & \cdots & x_{n-1}
\end{bmatrix},
\end{equation}

where $\tau$ is the time delay constant and $m$ is called the embedding dimension of the reconstructed phase space.

[Kennel et al., 1992] proposed an efficient method of finding the minimal sufficient embedding dimension, named as false nearest neighbors (FNN) procedure, in which the nearest neighbors of every point in a given dimension are found, and then checks are made to see if these points are still close neighbors in one higher dimension. To estimate the delay parameter, here we use the first minimum of the Mutual Information (MI) function [Abarbanel, 2012].

After finding the optimal $m$ and $\tau$, the input data and the output vector were designed by Eq. (<ref>) and Eq. (<ref>).

\begin{equation}\label{ytakens}
Y =
\begin{bmatrix}
\vdots\\
\end{bmatrix}
\begin{bmatrix}
\vdots\\
\end{bmatrix}
\end{equation}

After construction of the input and output matrix, we normalize the data by using min-max formula

\begin{equation}\label{2.4}
x_{\text{new}} = \frac{x_{\text{old}}-x_{\text{min}}}{x_{\text{max}}-x_{\text{min}}},
\end{equation}

to scale to the range $[0,1]$. Finally, the data is divided into the training set and the testing set.

In the first step of our proposed algorithm, the GSA parameters including the number of agents and maximum number of iterations are set. Then, GSA-SVR starts with a set of candidate solutions generated randomly within predetermined lower and upper bounds. In this case, each solution is a three-dimensional vector represented by ($C$, $\gamma$, $\epsilon$), where $C$, $\gamma$ and $\epsilon$ are the SVR parameters to be optimized. The objective function is equal to the Mean Square Error (MSE), Eq. <ref>, of the tested SVR model. A predetermined maximum number of iterations is used as a criteria to stop the algorithm. Figure <ref> shows flowchart of the complete the complete procedure.

A step-wise procedure of the proposed algorithm is described next:

Step 1:
Assign the parameters including the number of search agents and the maximum number of iterations. Set the iteration number, $t$, equal to zero.

Step 2:
Initialize the random solutions of search agents with

\begin{equation}\label{2.5}
S_i = Lb_i + (Ub_i-Lb_i)\cdot u
\end{equation}

and evaluate fitness function using Eq. (<ref>) and Eq. (<ref>) on the test data. Here, $Lb$ $(Ub)$ is lower (upper) bound and $i \in \{C, \gamma, \epsilon\}$, and $u$ is a uniform random number in the interval $(0,1)$.

Step 3:
Update the position of search agents for every dimension based on Eq. (<ref>) and set $t=t+1$.

Step 4:
If the maximum number of iterations is reached then the optimized parameters of SVR are selected, thus go to step 5; otherwise go back to step 3.

Step 5:
Use the SVR model with the optimal parameters ($C$, $\gamma$, $\epsilon$) for prediction.

rect = [draw = black, rectangle, fill=white!20, text width=10em, text centered, minimum height=2em, text= black]
elli = [draw, ellipse, fill=white!20, minimum height=2em, text= black]
circ = [draw, circle, fill=white!20, minimum width=8em, inner sep=10pt, text= black]
diam = [draw = black, diamond, fill=white!20,  minimum width=6em,  minimum height=2em, text width=6em, text height=0.05em, text badly centered, inner sep = 0pt, text= black]
line = [draw, -latex', text= black]

[node distance = 1.5cm]
[rect](step2)Initialize the three parameters $C$, $\gamma$ and $\epsilon$ randomly;
[rect, below of = step2] (step3) Train SVR;
[rect, left of = step3, node distance=6cm] (step4) Normalized data;
[rect, below of = step3, node distance=2cm]
(step5) Update the positions of agents according to the movement based on the operator in the GSA algorithm;
[diam, below of = step5, node distance=3cm] (step6) Has the number of iteration reached max iterations?;
[rect, rounded corners, below of = step6, node distance=2.5cm] (step7) Show the latest result;
[line, color=black] (step2) – (step3);
[line, color=black] (step3) – (step5);
[line, color=black] (step4) – (step3);
[line, color=black] (step5) – (step6);
[line, color=black] (step6) -| +(3,0) |- node [anchor=south] No (step3);
[line, color=black] (step6) – node [anchor = east] Yes (step7);

GSA-SVR procedure

§ EXPERIMENTAL RESULTS

In this section, a number of stocks are chosen to test the performance of our proposed GSA-SVR model. The proposed algorithm is compared to the other meta-heuristic algorithms, being used for parameter optimization of SVR, including Whale Optimization Algorithm based SVR (WOA-SVR), Salp Swarm Algorithm based SVR (SSA-SVR), Neural Network Algorithm based SVR (NNA-SVR), Firefly Algorithm based SVR (FA-SVR), Multi-Verse Optimizer based SVR (MVO-SVR), Moth-Flame Optimizer based SVR (MFO-SVR), Harris Hawks Optimization Algorithm based SVR (HHO-SVR), Grey Wolf Optimization Algorithm based SVR (GWO-SVR), Butterfly Optimization Algorithm based SVR (BOA-SVR), Biogeography-Based Optimization Algorithm based SVR (BBO-SVR) and Artificial Bee Colony Optimization Algorithm based SVR (ABC-SVR).

Stock market price prediction is regarded as one of the most challenging tasks of financial time series prediction. The difficulty of forecasting arises from the inherent non-linearity and nonstationarity of the stock market and financial time series.
Thus, daily closing stock market prices of three companies, namely Alibaba Group Holding Limited (BABA) (from 03/10/2016 to 01/10/2019), Tesla, Inc. (TSLA) (from 03/10/2016 to 01/10/2019) and Taiwan Semiconductor Manufacturing (TSM) Company Limited (from 03/10/2016 to 01/10/2019), were extracted from Yahoo Finance historical quotes. After finding the time delay, $\tau$, the embedding dimension, $m$ and reconstructing the phase space, $80\%$ of the data were used as the training set and the remaining were used as the testing set. All the predictions were based on one-step ahead prediction results and the computations were carried out in MATLAB R2019a environment using the LIBSVM Toolbox [Chang and Lin, 2011] on a laptop with an Intel(R) Core(TM) i3-3110M CPU @ 2.40GHz and 4 Gbytes memory.

In our work, Mean Squared Error (MSE) and Mean Absolute Percent Error (MAPE) were used in order to calculate the accuracy,

\begin{align}
\text{MSE} &= \frac{1}{N}\sum_{i=1}^{N}(y_i - f_i), \label{2.6}\\
\text{MAPE} &= \frac{1}{N}\sum_{i=1}^{N}\big|\frac{y_i - f_i}{y_i} \big|, \label{2.7}
\end{align}

where $y_i$ and $f_i$ denote the actual and predicted values for the $i$-th data point, respectively and $N$ is the number of forecasting days.

Since meta-heuristics algorithms use initial random population, we ran each algorithm several times to get the optimal answer. However, to increase the probability of finding the global optimum we used diversity in population and a sufficiently large number of iterations. In this study, the number of populations and the maximum number of iterations are selected to be 20 and 50 respectively. Also, the search space for both parameters $C$ and $\gamma$ were $[4^{-7},4^4]$ and the range for parameter $\epsilon$ was $[4^{-7}, 0.25]$.

All details of datasets including name, embedding dimension $m$ and time delay $\tau$  are shown in Table <ref>. The size of training datasets is equal to 530 and the size of testing datasets is equal to 133 for all of three datasets. For phase space reconstruction we used the recurrence plot and recurrence quantification analysis of MATLAB toolbox [Chen and Yang, 2012].

Actual and predicted values obttained by our model compared to eleven other methods for the three datasets TSM (Fig. (<ref>)), BABA (Fig. (<ref>)) and TSLA (Fig. (<ref>)) are illustrated in Fig. <ref> after de-normalization. Also, Table <ref> presents the optimal values for the three parameters $C, \gamma$ and $\epsilon$, as well as MSE, MAPE and the computing time for BABA testing dataset for all of SVR-based methods. The same results for TSLA and TSM testing datasets are shown in Tables <ref> and <ref>, respectively.

Prediction comparison for datasets a) TSM b) BABA and c) TSLA show that among the twelve algorithms, GSA-SVR performs as one of the best based on accuracy.

Time Consumption

Fig. <ref> and Fig. <ref> compare the MAPE and the MSE of the twelve methods. Also Fig. <ref> compares the cost time.

We now discuss results of our proposed algorithm in comparison with others. As shown in Fig. <ref>, the MAPE of GSA-SVR ranked 6th with a slight difference of 0.001 with the first rank, MFO-SVR. Even though MAPE accuracy of GSA-SVR is slightly below the other five methods MFO-SVR, MVO-SVR, GWO-SVR, ABC-SVR and FA-SVR, its computing time is significantly better than these methods except for GWO-SVR.

Bar plot of MAPEs

Bar plot of MSEs

Bar plot of Time Consumption

Fig. <ref> and Fig. <ref> compare the MAPE and the MSE bar time plots of the twelve methods. Also Fig. <ref> compare the cost time.

Also, Fig. <ref> illustrates that GSA-SVR, FA-SVR, ABC-SVR, MFO-SVR, GWO-SVR and SSA-SVR achieved the best MSE accuracy below 0.0012 in comparison with all the other methods. Thus it shows that the proposed algorithm is one of the best based on MSE error. Finally, Fig. <ref>, Fig. <ref> and Fig. <ref> respectively depict the bar plots of MAPE , MSE and cost time. Based on Fig. <ref>, the average time consumption of the GSA-SVR model ranked seventh but competitively close to the other methods. Also it is important to mention that although BBO-SVR, BOA-SVR, HHO-SVR and NNA-SVR algorithms are computationally less expensive, but their MSE and MAPE accuracy are very bad in comparision with GSA-SVR algorithm as it is shown in Fig. <ref> and Fig. <ref>.

Also to compare the predictive accuracy results of our method with others here we used Diebold-Mariano test [Diebold and Mariano, 2002]. Base on the test, the null hypothesis of equality of any two given methods at the $5\%$ confidence level is rejected if $|DM| > 1.96$, where DM is the test statistic of the Diebold-Mariano test calculated based on the corresponding squared-error residuals.

DM-values obtained by the Diebold-Mariano test on our three stocks is presented in Table <ref>. As shown in the table, there is a significant difference between our algorithm and BBO-SVR, HHO-SVR, MVO-SVR and NNA-SVR and in fact GSA-SVR is better than these algorithms and also there is no significance difference between the proposed algorithm and the rest of methods for BABA data.
For TSLA, the null hypothesis of equality is rejected for GSA-SVR and BBO-SVR, BOA-SVR, HHO-SVR and NNA-SVR and indeed, GSA-SVR based on forecasting accuracy performed better. For TSM, HHO-SVR, WOA-SVR have DM-test absolute value greater than 1.96, therefore there is only significant difference between these models and GSA-SVR.

To summarize, based on time efficiency, MSE and MAPE measures, we conclude that our GSA-SVR algorithm is capable to find the
optimal values of the SVR parameters and can yield promising results and also is one of the best models among the same other mata-heuristics based SVR methods studied.

§ CONCLUSION AND FUTURE RESEARCH

In support vector regression, parameters namely, penalty factor, $C$, RBF kernel function width parameter, $\gamma$ and radius of the epsilon tube, $\epsilon$ can change the performance of the algorithm considerably.Therefore, there is a need to optimize the parameters in a appropriate way. In this study, a novel hybrid method based on Support Vector Regression and Golden Sine algorithm is presented for selection of above mentioned parameters. The proposed method were tested on three financial time series of technology based companies, Alibaba Group Holding Limited, Tesla, Inc. and Taiwan Semiconductor Manufacturing Company Limited, using their daily closing stock market prices. For invalidation, the results are compared with eleven other meta-heuristics algorithms based SVR. According to the experimental results, GSA-SVR is capable of tuning the parameters efficiently in terms of computational time, MSE and MAPE errors.

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