Reinforcement of a plate weakened by multiple holes with several patches for different types of a plate-patch attachment

0.34 MB·English

by  A. Y. Zemlyanova

Reinforcement of a plate weakened by multiple holes
with several patches for different types of a plate-patch
attachment
2
A. Y. ZEMLYANOVA
1
0
Department of Mathematics, IAMCS, TexasA&M University
2 Mailstop 3368, TexasA&M University, College Station, TX 77843-3368 USA
n [e-mail]
a
J

1
3

] Abstract
V
Themostgeneralsituationofthereinforcementofaplatewithmultipleholes
C
by severalpatches is considered. There is no restrictionon the number and the
.
h locationofthepatches. Twotypesofthepatchattachmentareconsidered: only
t
a alongtheboundaryofthepatchorbothalongtheboundaryofthepatchandthe
m boundaries of the holes which this patch covers. The unattached boundaries of
[ the holes may be loaded with given in-plane stresses. The mechanical problem

is reduced to the system of singular integral equations which can be further
1
reducedtothesystemofFredholmequations. Anewnumericalprocedureforthe
v
6 solutionofthesystemofsingularintegralequationsisproposedinthispaper. It
6 canbeobservedthatthisprocedureprovidesanoticeableimprovementfromthe
6 one presentedin Zemlyanova[8],ZemlyanovaandSilvestrov[9], [10]andallows
6
to achieve a significantly better numerical convergencewith less computational
.
1 effort.
0
Keywords: Reinforcement, patch repair, complex potentials, singular integral
2
1 equations.
:
v
i
X
r 1. Introduction
a
Patch repair is one of the most common techniques used to strengthen thin
constructions containing imperfections such as cracks and holes. There is a
significant amount of scientific literature dedicated to the topic with different
types of the plate-patch attachment being considered. In particular, the plate
andthepatchescanbeattachedtoeachotherusingadhesiveappliedalongparts
of their surfaces (two-dimensional) or along some contours (one-dimensional),
or only at certain point by using rivets. The detailed review of the literature
on the bonded (two-dimensional) repair of the cracked plates can be seen in
[1], [3], [6] and many others. Reinforcement of plates with holes is studied less
extensively. Engels et al. [2] have considered a bonded repair of an anisotropic
1
Reinforcement of a plate weakened by multiple holes with several patches 2
plate with a hole. Some problems on the reinforcement of a cracked plate with
circularholes havebeen studied numerically in [4]. The patch repairof a single
hole in a thin plate reinforced by a single patch is considered for two different
types of the plate-patch attachment in [8], [9]. To the author’s knowledge, a
patchrepairofmultiple holesin a plate by multiple patchesofarbitraryshapes
has not been a subject of many studies with an exception of [10].
In this paper, a reinforcement of a plate with multiple holes with several
patches is considered for two types of the plate-patch attachment: only along
the boundary of the patch or along both the boundary of the patch and the
boundaries of the holes covered by the patch. It is assumed that the patch is
loaded with in-plane stresses applied to the free unattached boundaries of the
holesandatinfinitypointoftheplate. Thelocation,thequantityandtheshape
of the patches can be arbitrary with the only condition that the boundaries of
the patches and the holes are smooth and do not touch each other or intersect.
The complex analysis methods are used to solve the problem. First, the
Muskhelishvili’s formulas [5] are used to describe the stresses and the displace-
ments in the plate and the patches through the complex potentials in the plate
and the patches. The singular integral representations of the complex poten-
tials are used then to reformulate the problem as a system of singular integral
equations. These representationswerefirstproposedby Savruk[7]to study the
problems for cracked plates. The uniqueness of the solution of the resulting
system of the singular integral equations can be proved in a manner similar to
[8], [9], [10].
One of the main goals of this paper is to propose an alternative technique
for a numerical solution of the derived system of singular integral equations.
The method of mechanical quadratures applied for the solution of similar sys-
tems in [8], [9], [10] is difficult to implement for a large number of contours due
to decreasing precision. The idea behind the method proposed here is in the
approximation of the unknown functions by trigonometric polynomials. Sub-
stituting these approximations into the system of singular integral equations
reduces this system to the system of linear algebraic equations for the coeffi-
cients of the trigonometric polynomials. Numerical examples show that even
the systems of the relatively small order are sufficient to achieve a good accu-
racy of the results. By considering numerical examples it can be shown that
thismethoddemonstratesanoticeablybetterconvergence,largerindependence
from the symmetry of the construction and, hence, can be applied to a wider
range of practical problems.
2. Boundary conditions
Consider an infinite thin elastic plate S weakened by several holes with the
boundaries L , j = 1,...,n+r. Thin elastic patches S , k = 1,...,n+m,
j k
with the boundaries Γ are used to reinforce the plate S (fig. 1). It is assumed
k
that the patch S , k =1,...,n, coversthe hole L and is attached to the plate
k k
S both along the boundary of the hole L and the boundary of the patch Γ .
k k
Other patches S , k = n+1,...,n+m, are attached to the plate only along
k
Reinforcement of a plate weakened by multiple holes with several patches 3
L1 Ln+r
L
1
Lk
Lm+n
L
j+1
L
j
Figure1: AplateS withholesandattached patches Sk.
their boundaries Γ and can be located anywhere in the plate S, in particular,
k
they may cover some of the holes L , j = n+ 1,...,n +r. Given in-plane
j
stresses p (t) act on the free boundaries of the holes L , j = n+1,...,n+r.
j j
It is assumed that the lines L and Γ are smooth Lyapunovcurves and do not
j k
touch or intersect each other.
Theplateandthepatchesarehomogeneousandisotropic. Theirthicknesses,
shear moduli and Poisson ratios are given by h, µ, ν and h , µ , ν , k =
k k k
1,...,m+n, correspondingly. The principal in-plane stresses σ∞ and σ∞ are
1 2
applied at infinity of the plate and act in the directions constituting the angles
α and α+π/2 with the positive direction of the real axis.
Assumethatthepatchesandtheplatearejoinedperfectlyalongthejunction
lines. This means that the displacements are equal both in the patch and the
plate frombothsides ofthe junctionlines andthe stressequilibrium conditions
are satisfied on the junction lines. These conditions together with the given
stressesonthefreeboundariesoftheholesleadtothefollowingsetofboundary
conditions:
(u+iv)+(t)=(u+iv)+(t)=(u+iv)−(t), t∈L , k =1,...,n, (2.1)
k k k
h(σ +iτ )+(t)+h (σ +iτ )+(t)=h (σ +iτ )−(t), (2.2)
n n k n n k k n n k
t∈L , k =1,...,n,
k
(σ +iτ )+(t)=p (t), t∈L , k =n+1,...,n+r, (2.3)
n n k k k
(u+iv)+(t)=(u+iv)−(t)=(u+iv)+(t), t∈Γ , k =1,...,n+m, (2.4)
k k
h(σ +iτ )+(t)+h (σ +iτ )+(t)=h(σ +iτ )−(t), (2.5)
n n k n n k n n
t∈Γ , k =1,...,n+m,
k
Reinforcement of a plate weakened by multiple holes with several patches 4
where (u+iv)(t) is the vector of the displacements at a point t of the plate
S or the patch S , and σ and τ are the tensile and the shear components of
k n n
the stress vector acting on the tangent line to the curves L or Γ . Parameters
k k
without a subscript are related to the plate S; parameters with a subscript “k”
are related to the patch S . Here and henceforth, superscripts “+” and “−”
k
denote the limit values of the stresses, the displacements and other parameters
from the left-hand and the right-hand sides correspondingly of the lines L or
k
Γ . The directionof the curveL is chosento be clockwiseand ofthe curveΓ
k k k
counterclockwise (fig. 1).
3. Integral representations of complex potentials
ThestressesandthederivativesofthedisplacementsintheplateS orinthe
patch S (k =1,...,m+n) on any contour L′ lying in S or in S correspond-
k k
ingly can be obtained from the Muskhelishvili’s complex potentials Φ(z), Ψ(z)
by the following formulas [5]:
dt
(σ +iτ )(t)=Φ(t)+Φ(t)+ (tΦ′(t)+Ψ(t)), (3.6)
n n
dt
d dt
2µ (u+iv)(t)=κΦ(t)−Φ(t)− (tΦ′(t)+Ψ(t)), t∈L′.
dt dt
Forthe patchS , allthe parametersandthe functions inthese formulasshould
k
betakenwiththesubscript“k”. Inthecaseofplanestresstakeκ=(3−ν)/(1+
ν) and κ =(3−ν )/(1+ν ), k =1,...,m+n.
k k k
We will derive the integral representations of the complex potentials in the
spiritof[7]. Themainideaistodividethe“plate-patches”systemintoseparate
components and introduce the unknown functions describing the jumps of the
stresses or the derivatives of the displacements on each of the lines L and Γ .
j k
These unknown functions may be chosen in such a manner that some of the
conditions (2.1) - (2.5) are satisfied automatically.
First, consider the infinite plate S with the holes L which is subjected to
j
the given in-plane stresses at infinity and on the lines L , j =n+1,...,n+r.
j
Thestressesinthe plateS haveajumpdiscontinuityonthecurveγ =∪n+mΓ
k=1 k
lying inthe interiorofthe plate,andthe displacements arecontinuousonthese
curves:
q(t)= (σ +iτ )+(t)−(σ +iτ )−(t) /2, t∈γ. (3.7)
n n n n
Extend formally the (cid:0)plate S to the full complex plan(cid:1)e so that the stresses are
continuous through the curve L = ∪n+rL and the derivatives of the displace-
j=1 j
ments have a jump discontinuity along this line:
2µ d
g′(t)= (u+iv)+(t)−(u+iv)−(t) , t∈L. (3.8)
i(κ+1)dt
(cid:0) (cid:1)
Reinforcement of a plate weakened by multiple holes with several patches 5
Then the complex potentials Φ(z) and Ψ(z) can be taken in the form [7]:
n+r Q 1 g′(t)dt (κ+1)−1 q(t)dt
k
Φ(z)=Γ− + + ,
z−z 2π t−z πi t−z
k=n+1 k ZL Zγ
X
n+r κQ¯ 1 g′(t)dt t¯g′(t)dt
Ψ(z)=Γ′+ k + − (3.9)
z−z 2π t−z (t−z)2
k=n+1 k ZL !
X
(κ+1)−1 κq(t)dt t¯q(t)dt
+ − , z ∈S,
πi t−z (t−z)2
Zγ !
Γ=(σ∞+σ∞)/4, Γ′ =(σ∞−σ∞)e−2iα/2,
1 2 2 1
X +iY
k k
Q = , X +iY =−i p (t)dt,
k k k k
2π(1+κ)
ZLk
wherez (k =n+1,...,n+r)isthearbitrarilyfixedpointinsideofthecontour
k
L .
k
Formally extend each patch S , k = 1,...,n, to the full complex plane so
k
thatoutsideofthepatchallthedisplacementsandthestressesareequaltozero.
The displacements in the patch S are continuous through the line L ⊂ S ,
k k k
and the stresses have a jump discontinuity on this line:
q (t)= (σ +iτ )+(t)−(σ +iτ )−(t) /2, t∈L . (3.10)
k n n k n n k k
(cid:0) (cid:1)
Inthe complexplanethus obtainedfromthe patchS thejump ofthevectorof
k
the stresses (σ +iτ ) (t) acting on the tangent line to the contour Γ is equal
n n k k
to (σ +iτ )+(t), andthe jump ofthe vectorofthe displacements (u+iv)(t) is
n n k
equal to (u+iv)+(t):
k
2µ d
g′(t)= k (u+iv)+(t), t∈Γ . (3.11)
k i(κ +1)dt k k
k
Accordingtothecondition(2.5)andtheformula(3.7)wehave(σ +iτ )+(t)=
n n k
−2d−1q(t), t∈Γ , d =h /h. Therefore, the complex potentials Φ (z), Ψ (z)
k k k k k k
can be taken in the form [7]:
(κ +1)−1 q (t)dt 1 2id−1q(t) dt
Φ (z)= k k + g′(t)+ k , (3.12)
k πi t−z 2π k κ +1 t−z
ZLk ZΓk(cid:18) k (cid:19)
(κ +1)−1 κ q (t)dt t¯q (t)dt
k k k k
Ψ (z)= −
k πi t−z (t−z)2
ZLk !
1 2iκ q(t) dt 2id−1q(t) t¯dt
+ g′(t)+ k − g′(t)+ k ,
2π k d (κ +1) t−z k κ +1 (t−z)2
ZΓk" k k ! (cid:18) k (cid:19) #
z ∈S , k =1,...,n.
k
Reinforcement of a plate weakened by multiple holes with several patches 6
Observe that the representations (3.12) can be also used if the patch S
k
covers several holes and is attached both along its own boundary Γ and all
k
the boundaries of the holes which are covered by S . In this case the line L
k k
needstobeunderstoodastheunionofalltheboundariesoftheholeswhichare
coveredby S .
k
The same representations (3.12) are also valid for the patches S , k = n+
k
1,...,n+m, assuming that q (t)=0.
k
Hence, the stressed state of the “plate-patches” system is described by the
complex potentials (3.9), (3.12) which contain unknown functions g′(t), t ∈
L , j = 1,...,n + r; q (t), t ∈ L , j = 1,...,n, and q(t), g′(t), t ∈ Γ ,
j j j k k
k = 1,...,n+m. We will look for these functions in the class of functions
satisfying the Ho¨lder condition on the corresponding curves L and Γ . This
j k
choice guarantees the existence of all the principal and the limit values of the
integrals of the Cauchy type in the formulas (3.9), (3.12).
4. The system of singular integral equations
Tofindtheunknownfunctionsg′(t), q (t),q(t), g′(t)wehavetheconditions
k k
(2.1)-(2.4) and (3.11). Recall that the condition (2.5) has been used to derive
representations (3.12) and thus is satisfied automatically. Observe also that
the condition (3.11) appears from the way we extend each patch S to the full
k
complex plane (that is, it guarantees that the displacements and the stresses
outside of the patch S are equal to zero).
k
From the formulas (3.6) and the representations (3.9), (3.12), satisfying the
conditions(2.1)-(2.4)and(3.11),weobtainthesystemofsingularintegralequa-
tions on the closed curves L , j =1,...,n+r, and Γ , k =1,...,n+m, with
j k
the unknown functions g′(t), q (t), q(t), g′(t):
k k
µ dt n+r κQ Q¯ dt κQ t¯Q
k κΓ−Γ¯−Γ¯′ − j − j + j + j
µ  dt t−z t¯−z¯ dt t¯−z¯ (t¯−z¯ )2
j=n+1(cid:18) j j (cid:18) j j (cid:19)(cid:19)
X

i(κ+1) 1 κ 1 dt
+ g′(t)+ − g′(τ)dτ
2 2π τ −t τ¯−t¯dt
ZL(cid:18) (cid:19)
1 1 τ −t dt
+ − + g′(τ)dτ
2π τ¯−t¯ (τ¯−t¯)2dt
ZL(cid:18) (cid:19)
(κ+1)−1 κ κ dt
+ + q(τ)dτ
πi τ −t τ¯−t¯dt
Zγ(cid:18) (cid:19)
(κ+1)−1 1 τ −t dt
+ − q(τ)dτ
πi τ¯−t¯ (τ¯−t¯)2dt
Zγ(cid:18) (cid:19) (cid:21)
(κ +1)−1 κ κ dt
k k k
= + q (τ)dτ
πi τ −t τ¯−t¯dt k
ZLk(cid:18) (cid:19)
Reinforcement of a plate weakened by multiple holes with several patches 7
(κ +1)−1 1 τ −t dt
k
+ − q (τ)dτ
πi τ¯−t¯ (τ¯−t¯)2dt k
ZLk(cid:18) (cid:19)
1 κ 1 dt
+ k − g′(τ)dτ
2π τ −t τ¯−t¯dt k
ZΓk(cid:18) (cid:19)
1 1 τ −t dt
+ − + g′(τ)dτ
2π τ¯−t¯ (τ¯−t¯)2dt k
ZΓk(cid:18) (cid:19)
id−1 κ κ dt
+ k k + k q(τ)dτ
π(κ +1) τ −t τ¯−t¯dt
k ZΓk(cid:18) (cid:19)
id−1 1 τ −t dt
+ k − q(τ)dτ, t∈L , k =1,...,n; (4.13)
π(κ +1) τ¯−t¯ (τ¯−t¯)2dt k
k ZΓk(cid:18) (cid:19)
1 1 1 dt
2d q (t)+ + g′(τ)dτ
k k 2π τ −t τ¯−t¯dt
ZL(cid:18) (cid:19)
1 1 τ −t dt
+ − g′(τ)dτ
2π τ¯−t¯ (τ¯−t¯)2dt
ZL(cid:18) (cid:19)
(κ+1)−1 1 κ dt
+ − q(τ)dτ
πi τ −t τ¯−t¯dt
Zγ(cid:18) (cid:19)
(κ+1)−1 1 τ −t dt
+ − + q(τ)dτ
πi τ¯−t¯ (τ¯−t¯)2dt
Zγ(cid:18) (cid:19)
dt n+r Q dt tQ¯ κQ
=−2ReΓ−Γ¯′ + 2Re j − j + j ,
dt t−z dt (t¯−z¯ )2 t¯−z¯
j=n+1(cid:18) (cid:18) j(cid:19) (cid:18) j j(cid:19)(cid:19)
X
(4.14)
t∈L , k =1,...,n;
k
1|dt| 1 1 1 dt
g′(τ)dτ + + g′(τ)dτ
π dt 2π τ −t τ¯−t¯dt
ZLk ZL(cid:18) (cid:19)
1 1 τ −t dt
+ − g′(τ)dτ
2π τ¯−t¯ (τ¯−t¯)2dt
ZL(cid:18) (cid:19)
(κ+1)−1 1 κ dt
+ − q(τ)dτ
πi τ −t τ¯−t¯dt
Zγ(cid:18) (cid:19)
(κ+1)−1 1 τ −t dt
+ − + q(τ)dτ
πi τ¯−t¯ (τ¯−t¯)2dt
Zγ(cid:18) (cid:19)
dt n+r Q dt tQ¯ κQ
=−2ReΓ−Γ¯′ + 2Re j − j + j +p (t),
dt t−z dt (t¯−z¯ )2 t¯−z¯ k
j=n+1(cid:18) (cid:18) j(cid:19) (cid:18) j j(cid:19)(cid:19)
X
t∈L , k =n+1,...,n+r; (4.15)
k
Reinforcement of a plate weakened by multiple holes with several patches 8
µ dt n+r κQ Q¯ dt κQ t¯Q
k κΓ−Γ¯−Γ¯′ − j − j + j + j
µ  dt t−z t¯−z¯ dt t¯−z¯ (t¯−z¯ )2
j=n+1(cid:18) j j (cid:18) j j (cid:19)(cid:19)
X

1|dt| 1 κ 1 dt
+ q(τ)dτ + − g′(τ)dτ
π dt 2π τ −t τ¯−t¯dt
ZΓk ZL(cid:18) (cid:19)
1 1 τ −t dt
+ − + g′(τ)dτ
2π τ¯−t¯ (τ¯−t¯)2dt
ZL(cid:18) (cid:19)
(κ+1)−1 κ κ dt
+ + q(τ)dτ
πi τ −t τ¯−t¯dt
Zγ(cid:18) (cid:19)
(κ+1)−1 1 τ −t dt
+ − q(τ)dτ
πi τ¯−t¯ (τ¯−t¯)2dt
Zγ(cid:18) (cid:19) (cid:21)
i(κ +1) (κ +1)−1 κ κ dt
= k g′(t)+ k k + k q (τ)dτ
2 k πi τ −t τ¯−t¯dt k
ZLk(cid:18) (cid:19)
(κ +1)−1 1 τ −t dt
k
+ − q (τ)dτ
πi τ¯−t¯ (τ¯−t¯)2dt k
ZLk(cid:18) (cid:19)
1 κ 1 dt
+ k − g′(τ)dτ
2π τ −t τ¯−t¯dt k
ZΓk(cid:18) (cid:19)
1 1 τ −t dt
+ − + g′(τ)dτ
2π τ¯−t¯ (τ¯−t¯)2dt k
ZΓk(cid:18) (cid:19)
id−1 κ κ dt
+ k k + k q(τ)dτ
π(κ +1) τ −t τ¯−t¯dt
k ZΓk(cid:18) (cid:19)
id−1 1 τ −t dt
+ k − q(τ)dτ =i(κ +1)g′(t), (4.16)
π(κ +1) τ¯−t¯ (τ¯−t¯)2dt k k
k ZΓk(cid:18) (cid:19)
t∈Γ , k =1,...,n+m;
k
where q (t)=0, k =n+1,...,n+m, and d =h /h, k =1,...,n+m.
k k k
Observe, that the additional terms 1|dt| g′(τ)dτ and 1|dt| q(τ)dτ
π dt Lk π dt Γk
havebeenincludedintotheequations(4.15)and(4.16)correspondingly. Adding
R R
these terms does not change the conditions (2.1)-(2.4) and (3.11) and provides
that the following conditions are satisfied:
g′(τ)dτ =0, k =n+1,...,n+r, (4.17)
ZLk
q(τ)dτ =0, k=1,...,n+m. (4.18)
ZΓk
Fromthe physicalviewpointthe condition(4.17) means that the displacements
are single-valuedin the plate along the contoursL , k =n+1,...,n+r, while
k
Reinforcement of a plate weakened by multiple holes with several patches 9
the condition (4.18) providesthat the resultantforce applied to the patchS is
k
equal to zero [8], [10].
It can be shownthat the system (4.13)-(4.16) has a unique solution. This is
accomplished by reducing the system (4.13)-(4.16) to the system of Fredholm
equationsandprovingtheuniquenessofthesolutionofthemechanicalproblem.
Thishasbeenprovedindetailsforthecaseoftheplatewithoneholereinforced
byonepatchin[8]. Theproofforthegeneralcasecanbeobtainedusingsimilar
methods.
5. Numerical procedure
In this section we present an alternative technique for solving the system of
singularintegralequations(4.13)-(4.16). Themethodofmechanicalquadratures
presentedto study the problemsforcrackedplatesin [7]andadoptedforplates
with holes reinforced by patches in [8], [9], [10] provides a sufficient accuracy
with a reasonable computational time for the systems of similar type in the
cases when the number of contours L and Γ is small. However, the results
j k
become unsatisfactory both in the sense of accuracy and computational time if
a larger number of contours is considered. Another observed disadvantage of
the method of mechanical quadratures is that the results obtained using this
method tend to deteriorate significantly with the loss of the symmetry in the
construction.
To overcome these disadvantages the following approach is proposed and
showntobe computationallyeffective. First,parametrizeeachcurveL andΓ
j k
with a parameter θ ∈[0,2π]. Let τ =τ (θ) be a parametrizationof the curves
j j
L , and ξ = ξ (θ) be a parametrization of the curves Γ . Next, approximate
j k k k
theunknownfunctionsg′(t), q (t), t∈L ,andq(t), g′(t), t∈Γ ,bythepartial
k j k k
sums of the complex Fourier series by the parameter θ:
N
g′(t)= gmeimθ, t∈L , j =1,...,n+r;
0j j
m=−N
X
N
q (t)= qmeimθ, t∈L , j =1,...,n; (5.19)
j j j
m=−N
X
N
q(t)= qmeimθ, t∈Γ , j =1,...,n+m;
0j j
m=−N
X
N
g′(t)= gmeimθ, t∈Γ , j =1,...,n+m,
j j j
m=−N
X
wheregm, qm, qm,gm aretheunknowncomplexcoefficientsofthepartialsums
0j j 0j j
(5.19) which need to be found.
Reinforcement of a plate weakened by multiple holes with several patches 10
0.8
0.6
Γ Γ
0.4 1 2
0.2 L1 L2
y 0
−0.2
−0.4
−0.6
−0.8
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
Figure2: Aplatewithtwosquareholesreinforcedbytwosquarepatches.
Substituting the formulas (5.19) into the system (4.13)-(4.16) leads to the
following equations:
n+r N N
gmAm (θ)+ gmBm (θ)
0k j,k 0k j,k
!
k=1 m=−N m=−N
X X X
n+m N N
+ qmAm (θ)+ qmBm (θ)
0k j,k+n+r 0k j,k+n+r
!
k=1 m=−N m=−N
X X X
n N N
+ qmAm (θ)+ qmBm (θ) (5.20)
k j,k+2n+m+r k j,k+2n+m+r
!
k=1 m=−N m=−N
X X X
n+m N N
+ gmAm (θ)+ gmBm (θ) =f (θ),
k j,k+3n+m+r k j,k+3n+m+r j
!
k=1 m=−N m=−N
X X X
θ ∈[0,2π], j =1,...,4n+2m+r,
where N is the degree of the trigonometric polynomials (5.19), and the func-
tions Am (θ), Bm (θ) and f (θ) can be obtained from the system (4.13)-(4.16).
j,k j,k j
The integrals in these functions can be computed numerically by using any
appropriate method.
The equations (5.20), in general, can not be satisfied for all the values of
the parameter θ due to the approximation of the unknown functions by the
truncated Fourier series (5.19). Instead, we assume that the equations (5.20)
are satisfied only in the finite number of points θ = π(2s − 1)/(2N + 1),
s
s = 1,...,2N +1. This transforms (5.20) into the system of linear algebraic
equations with real coefficients of the order 2(2N+1)(4n+2m+r). Thus, the
approximate solution of the system of singular integral equations (4.13)-(4.16)
can be obtained by solving the system of linear algebraic equations.
While the precise investigation of the convergence of the solution of the
resulting system of linear algebraic equations to the solution of the system of
singular integral equations (4.13)-(4.16) is outside of the scope of this paper,
the numerical results obtained by using this method and also comparison with