greculumialbi. 112KB Jun 04 2011 12:08:43 AM
ACTA UNIVERSITATIS APULENSIS
No 11/2006
Proceedings of the International Conference on Theory and Application of
Mathematics and Informatics ICTAMI 2005 - Alba Iulia, Romania
THE BOUNDARY ELEMENT METHOD WITH LINEAR
BOUNDARY ELEMENTS FOR THE STOKES FLOW
Grecu Luminit
¸a
Abstract. The aim of the paper is to apply the boundary element method
to solve the problem of the Stokes flow, in fact to solve the second step in applying this method, the system of singular boundary integral equations the
problem is reduced at, using linear isoparametric boundary elements of Lagrangean type.
2000 Mathematics Subject Classification:74S15
1.Introduction
The boundary integral method (BEM) is a powerful numerical technique
used to solve boundary value problems for systems of partial differential equations. It has the ability to reduce the problem dimension by one, and it has two
major steps. The first step is to formulate the problem in terms of a boundary
integral equation, or a system of boundary integral equations and the second
step is to solve the boundary integrals obtained (see[1]).
We apply the boundary element method for solving the problem of the
steady slow viscous flow of an incompressible fluid over an obstacle, a fluid
that has the uniform subsonic velocity u¯0 at great distances. We consider that
the uniform steady motion is perturbed by the presence of a fixed body of a
known boundary, noted Σ, assumed to be smooth and closed. We want to find
out the perturbed motion, and the fluid action on the body.
A boundary integral representation for the velocity is obtained in [2] ,using
an indirect technique, so the fundamental solution of the mathematical model
of the Stokes flow, and an integral representation of the pressure is obtained
in [3]. The problem is reduced to a system of singular boundary integral
equations. Because the singular integrals that appear exist only if we assume
that the unknown function satisfies a h˝older condition on the boundary, so it
is at least continuous on it, the system obtained is solved in this paper using
linear isoparametric boundary elements of Lagrangean type.
139
L. Grecu - The boundary element method with linear boundary ...
Both, the geometry and the unknown, have local a linear behavior, and
the solution obtained is continuous on the boundary. After the discretization
of the boundary, we choose a linear model for approximating the unknown
on each of the elements involved. We obtain a linear algebraic system whose
solutions are the unknowns on the nodes chosen for the discretization.After
solving this system we can find the velocity in any point of the boundary and
everywhere in the fluid domain. Using this technique it can be also obtained
the pressure on the boundary and the fluid action over the obstacle.
2.The system of boundary integral equations
The mathematical model of the problem is described by the following equations for the perturbed motion ( ν is the coefficient of the cinematic viscosity)(see [4]):
(1)
∇p − ν∆u = 0, ∇ · u = 0
and the boundary conditions: u1 (x) = ν0 , x ∈ Σ, u2 (x) = u3 (x) = 0, x ∈
Σ, lim∞ (u, p) = 0
Assimilating the body with a continuous distribution of sources on the
boundary, having an unknown intensity f , and assuming that f satisfies a
holder condition
in any of the arguments we obtain a boundary representation for ui ξ 0 , Q0 ξ 0 being an arbitrary point situated on Σ. Introducing
the boundary conditions, the problem can be reduced to the following system
of boundary integral equations (see [2]):
1
8πν
′
Z Z
Σ
fi (ξ1 , ξ2 , ξ3 )
1
dσ+
0
r
8πν
′
Z Z
ξ−ξ
0
i
Σ
ξ−ξ
0
(r0 )3
· f (ξ1 , ξ2 , ξ3 )
dσ = −νi0 ,
(2)
for i = 1, 2, 3,
h
2
2
2
where ν20 = ν30 = 0 , r0 = (ξ1 − ξ10 ) + (ξ2 − ξ20 ) + (ξ3 − ξ30 )
For the pressure on the boundary we obtain(see [3]):
p ξ0
i1
2
0
1
1 ′Z Z f ξ ξ −ξ
dσ
−
=
f ξ 0 · n0 .
0 )3
4π
2
(r
Σ
140
.
(3)
L. Grecu - The boundary element method with linear boundary ...
The sign ”′ ” denotes the principal value in Cauchy sense of an integral.
3.The discrete form of the equations
A collocation method is used for example in [7] for solving the system of
integral equations to determine the perturbation velocity and pressure, but
the unknown is there constant on every boundary element, so it is not even
continuous on the boundary, as the representation asked. In the boundary
element approach used herein, for solving the system of integral equations (2),
the body surface, Σ , is approximated by N triangles, noted Tj , j = 1, N ,
with straight lines; the extremes xi , i = 1, M of these triangles being on Σ.
The system of integral equations becomes:
N Z Z
fi (ξ1 , ξ2 , ξ3 )
1 X
dσ +
8πν j=1
r0
(4)
Tj
+
1
8πν
N Z Z
X
j=1
ξ−ξ
0
i
ξ−ξ
Tj
0
(r0 )3
· f (ξ1 , ξ2 , ξ3 )
dσ = −vi0 , i = 1, 2, 3. (5)
0
Considering ξ = xk , k ∈ {1, 2, ..., M }, we have:
N Z Z
fi (ξ1 , ξ2 , ξ3 )
1 X
dσ +
8πν j=1
rk
(6)
Tj
+
1
8πν
N Z Z
X
j=1
Tj
ξ − xk
i
ξ − xk · f (ξ1 , ξ2 , ξ3 )
(rk )3
dσ = −vi0 , i = 1, 3 (7)
We have to calculate two types of integrals on Tj , with, and without singularities. If xk is a corner point of the triangle Tj , the integral calculated on Tj
has a singularity, otherwise it doesn’t. We calculate the integrals using a local
system of coordinates. We consider that the corner points (nodes) of an arbitrary triangle Tj are noted
with j1 , j2 , j3 . Denoting by λ1 , λ2, λ3 the intrinsic
triangular coordinates λi ∈ [0, 1] , i = 1, 3 , λ1 + λ2 + λ3 = 1 ,we have for an
interior point of the triangle the relation:
ξ = xj1 + (xj2 − xj1 ) λ2 + (xj3 − xj1 ) λ3 .
141
L. Grecu - The boundary element method with linear boundary ...
Using
the parametric representation: λ2 = r cos θ, λ3 = r sin θ, θ ∈
i
0, π2 , r ∈ [0, ρ] , ρ (cos θ + sin θ) = 1we get: ξ = xj1 + (xj2 − xj3 ) r cos θ +
(xj3 − xj1 ) r sin θ. We also have da = 2Sdλ1 dλ2 = 2Srdrdθ, where S is the
area of the triangle. For describing the behavior of the unknown f , we use a
linear model:
h
f =f
j1
+ f
j2
−f
j1
r cos θ + f
j3
−f
j1
r sin θ
(8)
j
where f l , l = 1, 3 is the value of the unknown f for the node jl .
The system (5) becomes:
N
P
j=1
Ajki fij1 +
N
P
j=1
j
fij2 − fij1 +
Bki
j
+E k · f
j2
−f
j1
+
N
P
j=1
N
X
j=1
j
Fk · f
j3
−f
j1
N
P
j
j=1
= −νi0
j
fij3 − fij1 +
Cki
j1
Dk · f +
(9)
The coefficients have the following expressions:
Ajki =
1 Z Z 1
1 Z Z r cos θ
j
=
dσ,
B
dσ,
ki
8πν
rk
8πν
rk
Tj
j
=
Cki
j
Ek =
j
Fk
1
8πν
Z Z
1
8πν
Z Z
Tj
Tj
1
r sin θ
j
dσ, Dk =
k
r
8πν
Z Z
ξi − xki
Tj
ξ − xk
(rk )3
dσ,
(11)
ξi − xki
ξ − xk r cos θ
dσ,
(12)
ξi − xki
ξ − xk r sin θ
dσ
(13)
Tj
1 Z Z
=
8πν
(10)
Tj
(rk )3
(rk )3
For evaluating the system’ s coefficients we consider the two types of integrals that appear: the nonsingular and the singular ones. First we evaluate
the nonsingular integrals, so we consider the triangles that have no corner
point in xk . Denoting by
ej [θ] =
(xj2
− xj1 ) cos θ + (xj3
− xj1 ) sin θ, ej [θ] =
√
ej1 , ej2 , ej3 ,we get: rk =
ξˆ − xk
=
xj1 − xk + rej (θ)
= ar2 + 2br + c,
2
where a = kejk (θ)k2 , b = xj1 − xk · ej (θ) , c =
xj1 − xk
142
L. Grecu - The boundary element method with linear boundary ...
We also have the relations: da = 2Sj dλ1 dλ2 = 2Sj rdrdθ, where Sj is the
π
area of Tj , j = 1, N . We obtain:
Ajki
=
π
Cjki
=
Sj
4πν
R2
0
Sj
4πν
R2
0
π
J1 (θ) dθ,
j
Bki
=
Sj
4πν
R2
0
J2 (θ) cos θdθ
J2 (θ) sin θdθ,
π
j
Dk
Z2
Sj j1
j2
j1
k
I1 (θ) dθ +
=
x − xi
x −x
4πν i
0
π
2
+
π
2
Sj h 21 i Z
Sj h 31 i Z
qijk
q
cos θI2 (θ) dθ +
sin θI2 (θ) dθ +
4πν
4πν ijk
0
0
π
2
Sj h 23 i Z
sin θ cos θI3 (θ) dθ +
q
+
4πν ijk
0
π
i Z2
Sj h j2
j1
j3
j1
+
xi − xi
x −x
cos2 θI3 (θ) dθ +
4πν
0
π
i Z2
Sj h j3
j1
j3
j1
sin2 θI3 (θ) dθ
xi − xi
x −x
+
4πν
0
π
j
Ek
Z2
Sj j1
j2
j1
k
=
x −x
cos θI2 (θ) dθ +
x − xi
4πν i
0
π
2
+
π
2
Sj h 21 i Z
Sj h 31 i Z
cos2 θI3 (θ) dθ +
sin θ cos θI3 (θ) dθ +
qijk
q
4πν
4πν ijk
0
0
π
2
+
Sj h 23 i Z
sin θ cos2 θI4 (θ) dθ +
qijk
4πν
0
π
+
Sj
4πν
h
xji 2 − xji 1
xj3 − xj1
i Z2
cos3 θI4 (θ) dθ +
0
π
2
i Z
Sj h j3
j1
j3
j1
+
sin2 θ cos θI4 (θ) dθ
xi − xi
x −x
4πν
0
143
L. Grecu - The boundary element method with linear boundary ...
π
j
Fk
Z2
Sj j1
j2
j1
k
sin θI2 (θ) dθ +
=
x − xi
x −x
4πν i
0
π
2
+
π
2
Sj h 21 i Z
Sj h 31 i Z
qijk
q
sin θ cos θI3 (θ) dθ +
sin2 θI3 (θ) dθ +
4πν
4πν ijk
0
0
π
2
Sj h 23 i Z
sin2 θ cos θI4 (θ) dθ +
qijk
+
4πν
0
π
+
Sj
4πν
h
xji 2 − xji 1
xj3 − xj1
i Z2
cos2 θ sin θI4 (θ) dθ +
0
π
i Z2
Sj h j3
j1
j3
j1
sin3 θI4 (θ) dθ
xi − xi
+
x −x
4πν
21
qijk
=
xji 1
−
xki
31
qijk
= xji 1 − xki
23
qijk
= xji 2 − xji 1
j2
x −x
j1
+
xji 2
−
0
xji 1 xj1
xj3 − xj1 + xji 3 − xji 1
xj3 − xj1 + xji 2 − xji 1
− xk ,
xj1 − xk ,
xj2 − xj1
The coefficients noted J1 , J2 , J3 , I1 , I2 , I3 , I4 can be computed analytically
and they have the expressions:
√
√
J1 (θ) = a1 aρ2 + 2bρ + c − ac − ab J.
J2 (θ) =
J3 (θ) =
aρ−3b √
aρ2
2a2
+ 2bρ + c +
3b √
c
2a2
+
3b2 −ac
J.
2a2
2a2 ρ2 − 5abρ + 15b2 − 4ac q 2
15b2 − 4ac √
5b3 − 3bc
aρ
+
2bρ
+
c
−
c
−
J.
6a3
6a3
2a3
where
1
1
aρ + b
b
√ arcsin √
J = √ arcsin √
−
a
a
ac − b2
ac − b2
144
L. Grecu - The boundary element method with linear boundary ...
√
√
b
b + aρ + aρ2 + 2bρ + c
c
1q 2
√
I1 (θ) =
aρ + 2bρ + c −
− √ ln
.
a
a
a a
b + ac
√
I2 (θ) =
c
δ
−√
2
1
bρ + c
ρ2
b
√
+ I2 − I4
,
I
(θ)
=
−
3
2
2
a
aρ + 2bρ + c δ
a aρ + 2bρ + c a
q
√
a (aρ2 + 2bρ + c) + aρ + b
(b2 − δ) ρ + bc
b c
1
√
−
I4 (θ) = √ 2
+ √ ln
(14)
aδ
a a
b + ac
aδ aρ + 2bρ + c
We evaluate the singular integrals. Considering now that the triangle,
noted Tk , has a corner point in xk , which we take as node number one after
a renotation, we calculate the singular integrals occurring in (10) using the
k
following relations: ξ = xk + xj1 − xk r cos θ + xj2 − xk r sin θ , f = f +
f
j2
−f
k
r cos θ + f
j3
−f
k
r sin θ. Denoting by ejk [θ] = xj2 − xk cos θ +
xj3 − xk sin θ, we get rk =
ξ − xk
= krejk (θ)k = r kejk (θ)k, and the
coefficients have in this case the following expressions:
π
Sj
4πν
R2
Bjki =
Sj
4πν
R R
cos θ ρ2 (θ)
dθ,
kejk (θ)k 2
Cjki =
Sj
4πν
R R
sin θ ρ2 (θ)
dθ
kejk (θ)k 2
Ajki
=
0
1
kejk (θ)k
Tj
Tj
qijk = xji 2 − xki
j
Dk
+
Sj
4πν
=
h
Sj
4πν
π
dθ,
R2
[qijk ] sin θ cos θ3 dθ
0 kejk (θ)k
xji 3 − xki
xj3 − xk + xji 2 − xki
xj3 − xk
π
2
i R
0
+
Sj
4πν
h
xji 2
sin2 θ
kejk (θ)k
3
dθ
π
j
Ek
=
Sj
4πν
[qijk ]
R2
sin θ cos2 θ ρ2 (θ)
dθ+
3
2
0 kejk (θ)k
145
−
xj2 − xk .
xki
j3
x −x
k
i Rπ2
cos2 θ
3 dθ
e
0 k jk (θ)k
+
L. Grecu - The boundary element method with linear boundary ...
π
i Z2 cos3 θ ρ2 (θ)
Sj h j2
xi − xki xj3 − xk
+
dθ +
3
4πν
2
ke
(θ)k
jk
0
π
+
Sj
4πν
h
xji 3 − xki
xj3 − x
i Z2 sin2 θ cos θ ρ2 (θ)
k
0
kejk (θ)k3
2
dθ
π
j
Fk
=
Sj
4πν
[qijk ]
R2
sin2 θ cos θ ρ2 (θ)
dθ+
3
2
0 kejk (θ)k
π
+
Sj
4πν
h
xji 2 − xki
xj3 − x
i Z2 cos2 θ sin θ ρ2 (θ)
k
0
π
kejk (θ)k3
2
dθ +
i Z2 sin3 θ ρ2 (θ)
Sj h j3
k
j3
k
xi − xi x − x
dθ
+
3
4πν
2
ke
(θ)k
jk
0
(15)
(16)
As shown in the previous paragraphs all the system’ s coefficients depend
only on the coordinates of the nodes chosen for the boundary discretization,
and on some simple integrals that are one-dimensional and can be evaluated
with a math soft or with the usual rules of the numerical integration. After some manipulation we obtain a linear system of 3M equations with 3
M unknowns,the scalar components of the M nodal values of the unknown
j
functionf , f f1j , f2j , f3j , j = 1, M , having the following form:
M
X
j=1
Aji fij +
M
X
j=1
j
j
B i · f = −νi0 , i = 1, 3,
(17)
where Aji is the coefficient of fij , i = 1, 3 , j = 1, M obtained as a sum of
all the contributions that exist.After solving the system we may compute the
velocity for the M nodes chosen for the discretization of the boundary, and
everywhere on it, and also in the fluid region, and further the pressure on the
boundary and in the fluid region.
146
L. Grecu - The boundary element method with linear boundary ...
References
[1] Brebbia, C.A., Telles, J.C.F., Wobel, L.C.: Boundary Element Theory
and Application in Engineering, Springer-Verlag, Berlin,1984
[2] Grecu, L.: A new boundary element approach for the Stokes flow, Proceedings of ”microCAD 2005” International Scientific Conference, Section G,
63-68
[3] Grecu, L.: A boundary element approach for evaluating the fluid action over the obstacle for the Stokes flow, Proceedings of ”microCAD 2005”
International Scientific Conference, Section G, 69-72
[4] Dragos, L.: Principiile mecanicii mediilor continue, Ed.Tehnica, Bucuresti,1983
[5] Dragos, L.: Metode Matematice in Aerodinamica (Mathematical Methods in Aerodinamics), Editura Academiei Romane, Bucuresti, 2000
[6] Lifanov, I.K.: Singular integral equations and discrete vortices, VSP,
Utrecht, TheNetherlands, 1996
[7] Petrila,T.,Kohr,M.: Some boundary element techniques for Stokes flows
(PartI), Rev. D’Anallyse Numerique et de Theorie de L’Approximation 21(1992),
167-182.
Grecu Luminit¸a
Department of Applied Sciences
University of Craiova, Faculty of Engineering and Management of the Technological Systems, Drobeta Turnu Severin Address: C˘alug˘areni no.1 Dr. Tr.
Severin, cod 220037
email: luminita.grecu@imst.ro; lumigrecu@hotmail.com
147
No 11/2006
Proceedings of the International Conference on Theory and Application of
Mathematics and Informatics ICTAMI 2005 - Alba Iulia, Romania
THE BOUNDARY ELEMENT METHOD WITH LINEAR
BOUNDARY ELEMENTS FOR THE STOKES FLOW
Grecu Luminit
¸a
Abstract. The aim of the paper is to apply the boundary element method
to solve the problem of the Stokes flow, in fact to solve the second step in applying this method, the system of singular boundary integral equations the
problem is reduced at, using linear isoparametric boundary elements of Lagrangean type.
2000 Mathematics Subject Classification:74S15
1.Introduction
The boundary integral method (BEM) is a powerful numerical technique
used to solve boundary value problems for systems of partial differential equations. It has the ability to reduce the problem dimension by one, and it has two
major steps. The first step is to formulate the problem in terms of a boundary
integral equation, or a system of boundary integral equations and the second
step is to solve the boundary integrals obtained (see[1]).
We apply the boundary element method for solving the problem of the
steady slow viscous flow of an incompressible fluid over an obstacle, a fluid
that has the uniform subsonic velocity u¯0 at great distances. We consider that
the uniform steady motion is perturbed by the presence of a fixed body of a
known boundary, noted Σ, assumed to be smooth and closed. We want to find
out the perturbed motion, and the fluid action on the body.
A boundary integral representation for the velocity is obtained in [2] ,using
an indirect technique, so the fundamental solution of the mathematical model
of the Stokes flow, and an integral representation of the pressure is obtained
in [3]. The problem is reduced to a system of singular boundary integral
equations. Because the singular integrals that appear exist only if we assume
that the unknown function satisfies a h˝older condition on the boundary, so it
is at least continuous on it, the system obtained is solved in this paper using
linear isoparametric boundary elements of Lagrangean type.
139
L. Grecu - The boundary element method with linear boundary ...
Both, the geometry and the unknown, have local a linear behavior, and
the solution obtained is continuous on the boundary. After the discretization
of the boundary, we choose a linear model for approximating the unknown
on each of the elements involved. We obtain a linear algebraic system whose
solutions are the unknowns on the nodes chosen for the discretization.After
solving this system we can find the velocity in any point of the boundary and
everywhere in the fluid domain. Using this technique it can be also obtained
the pressure on the boundary and the fluid action over the obstacle.
2.The system of boundary integral equations
The mathematical model of the problem is described by the following equations for the perturbed motion ( ν is the coefficient of the cinematic viscosity)(see [4]):
(1)
∇p − ν∆u = 0, ∇ · u = 0
and the boundary conditions: u1 (x) = ν0 , x ∈ Σ, u2 (x) = u3 (x) = 0, x ∈
Σ, lim∞ (u, p) = 0
Assimilating the body with a continuous distribution of sources on the
boundary, having an unknown intensity f , and assuming that f satisfies a
holder condition
in any of the arguments we obtain a boundary representation for ui ξ 0 , Q0 ξ 0 being an arbitrary point situated on Σ. Introducing
the boundary conditions, the problem can be reduced to the following system
of boundary integral equations (see [2]):
1
8πν
′
Z Z
Σ
fi (ξ1 , ξ2 , ξ3 )
1
dσ+
0
r
8πν
′
Z Z
ξ−ξ
0
i
Σ
ξ−ξ
0
(r0 )3
· f (ξ1 , ξ2 , ξ3 )
dσ = −νi0 ,
(2)
for i = 1, 2, 3,
h
2
2
2
where ν20 = ν30 = 0 , r0 = (ξ1 − ξ10 ) + (ξ2 − ξ20 ) + (ξ3 − ξ30 )
For the pressure on the boundary we obtain(see [3]):
p ξ0
i1
2
0
1
1 ′Z Z f ξ ξ −ξ
dσ
−
=
f ξ 0 · n0 .
0 )3
4π
2
(r
Σ
140
.
(3)
L. Grecu - The boundary element method with linear boundary ...
The sign ”′ ” denotes the principal value in Cauchy sense of an integral.
3.The discrete form of the equations
A collocation method is used for example in [7] for solving the system of
integral equations to determine the perturbation velocity and pressure, but
the unknown is there constant on every boundary element, so it is not even
continuous on the boundary, as the representation asked. In the boundary
element approach used herein, for solving the system of integral equations (2),
the body surface, Σ , is approximated by N triangles, noted Tj , j = 1, N ,
with straight lines; the extremes xi , i = 1, M of these triangles being on Σ.
The system of integral equations becomes:
N Z Z
fi (ξ1 , ξ2 , ξ3 )
1 X
dσ +
8πν j=1
r0
(4)
Tj
+
1
8πν
N Z Z
X
j=1
ξ−ξ
0
i
ξ−ξ
Tj
0
(r0 )3
· f (ξ1 , ξ2 , ξ3 )
dσ = −vi0 , i = 1, 2, 3. (5)
0
Considering ξ = xk , k ∈ {1, 2, ..., M }, we have:
N Z Z
fi (ξ1 , ξ2 , ξ3 )
1 X
dσ +
8πν j=1
rk
(6)
Tj
+
1
8πν
N Z Z
X
j=1
Tj
ξ − xk
i
ξ − xk · f (ξ1 , ξ2 , ξ3 )
(rk )3
dσ = −vi0 , i = 1, 3 (7)
We have to calculate two types of integrals on Tj , with, and without singularities. If xk is a corner point of the triangle Tj , the integral calculated on Tj
has a singularity, otherwise it doesn’t. We calculate the integrals using a local
system of coordinates. We consider that the corner points (nodes) of an arbitrary triangle Tj are noted
with j1 , j2 , j3 . Denoting by λ1 , λ2, λ3 the intrinsic
triangular coordinates λi ∈ [0, 1] , i = 1, 3 , λ1 + λ2 + λ3 = 1 ,we have for an
interior point of the triangle the relation:
ξ = xj1 + (xj2 − xj1 ) λ2 + (xj3 − xj1 ) λ3 .
141
L. Grecu - The boundary element method with linear boundary ...
Using
the parametric representation: λ2 = r cos θ, λ3 = r sin θ, θ ∈
i
0, π2 , r ∈ [0, ρ] , ρ (cos θ + sin θ) = 1we get: ξ = xj1 + (xj2 − xj3 ) r cos θ +
(xj3 − xj1 ) r sin θ. We also have da = 2Sdλ1 dλ2 = 2Srdrdθ, where S is the
area of the triangle. For describing the behavior of the unknown f , we use a
linear model:
h
f =f
j1
+ f
j2
−f
j1
r cos θ + f
j3
−f
j1
r sin θ
(8)
j
where f l , l = 1, 3 is the value of the unknown f for the node jl .
The system (5) becomes:
N
P
j=1
Ajki fij1 +
N
P
j=1
j
fij2 − fij1 +
Bki
j
+E k · f
j2
−f
j1
+
N
P
j=1
N
X
j=1
j
Fk · f
j3
−f
j1
N
P
j
j=1
= −νi0
j
fij3 − fij1 +
Cki
j1
Dk · f +
(9)
The coefficients have the following expressions:
Ajki =
1 Z Z 1
1 Z Z r cos θ
j
=
dσ,
B
dσ,
ki
8πν
rk
8πν
rk
Tj
j
=
Cki
j
Ek =
j
Fk
1
8πν
Z Z
1
8πν
Z Z
Tj
Tj
1
r sin θ
j
dσ, Dk =
k
r
8πν
Z Z
ξi − xki
Tj
ξ − xk
(rk )3
dσ,
(11)
ξi − xki
ξ − xk r cos θ
dσ,
(12)
ξi − xki
ξ − xk r sin θ
dσ
(13)
Tj
1 Z Z
=
8πν
(10)
Tj
(rk )3
(rk )3
For evaluating the system’ s coefficients we consider the two types of integrals that appear: the nonsingular and the singular ones. First we evaluate
the nonsingular integrals, so we consider the triangles that have no corner
point in xk . Denoting by
ej [θ] =
(xj2
− xj1 ) cos θ + (xj3
− xj1 ) sin θ, ej [θ] =
√
ej1 , ej2 , ej3 ,we get: rk =
ξˆ − xk
=
xj1 − xk + rej (θ)
= ar2 + 2br + c,
2
where a = kejk (θ)k2 , b = xj1 − xk · ej (θ) , c =
xj1 − xk
142
L. Grecu - The boundary element method with linear boundary ...
We also have the relations: da = 2Sj dλ1 dλ2 = 2Sj rdrdθ, where Sj is the
π
area of Tj , j = 1, N . We obtain:
Ajki
=
π
Cjki
=
Sj
4πν
R2
0
Sj
4πν
R2
0
π
J1 (θ) dθ,
j
Bki
=
Sj
4πν
R2
0
J2 (θ) cos θdθ
J2 (θ) sin θdθ,
π
j
Dk
Z2
Sj j1
j2
j1
k
I1 (θ) dθ +
=
x − xi
x −x
4πν i
0
π
2
+
π
2
Sj h 21 i Z
Sj h 31 i Z
qijk
q
cos θI2 (θ) dθ +
sin θI2 (θ) dθ +
4πν
4πν ijk
0
0
π
2
Sj h 23 i Z
sin θ cos θI3 (θ) dθ +
q
+
4πν ijk
0
π
i Z2
Sj h j2
j1
j3
j1
+
xi − xi
x −x
cos2 θI3 (θ) dθ +
4πν
0
π
i Z2
Sj h j3
j1
j3
j1
sin2 θI3 (θ) dθ
xi − xi
x −x
+
4πν
0
π
j
Ek
Z2
Sj j1
j2
j1
k
=
x −x
cos θI2 (θ) dθ +
x − xi
4πν i
0
π
2
+
π
2
Sj h 21 i Z
Sj h 31 i Z
cos2 θI3 (θ) dθ +
sin θ cos θI3 (θ) dθ +
qijk
q
4πν
4πν ijk
0
0
π
2
+
Sj h 23 i Z
sin θ cos2 θI4 (θ) dθ +
qijk
4πν
0
π
+
Sj
4πν
h
xji 2 − xji 1
xj3 − xj1
i Z2
cos3 θI4 (θ) dθ +
0
π
2
i Z
Sj h j3
j1
j3
j1
+
sin2 θ cos θI4 (θ) dθ
xi − xi
x −x
4πν
0
143
L. Grecu - The boundary element method with linear boundary ...
π
j
Fk
Z2
Sj j1
j2
j1
k
sin θI2 (θ) dθ +
=
x − xi
x −x
4πν i
0
π
2
+
π
2
Sj h 21 i Z
Sj h 31 i Z
qijk
q
sin θ cos θI3 (θ) dθ +
sin2 θI3 (θ) dθ +
4πν
4πν ijk
0
0
π
2
Sj h 23 i Z
sin2 θ cos θI4 (θ) dθ +
qijk
+
4πν
0
π
+
Sj
4πν
h
xji 2 − xji 1
xj3 − xj1
i Z2
cos2 θ sin θI4 (θ) dθ +
0
π
i Z2
Sj h j3
j1
j3
j1
sin3 θI4 (θ) dθ
xi − xi
+
x −x
4πν
21
qijk
=
xji 1
−
xki
31
qijk
= xji 1 − xki
23
qijk
= xji 2 − xji 1
j2
x −x
j1
+
xji 2
−
0
xji 1 xj1
xj3 − xj1 + xji 3 − xji 1
xj3 − xj1 + xji 2 − xji 1
− xk ,
xj1 − xk ,
xj2 − xj1
The coefficients noted J1 , J2 , J3 , I1 , I2 , I3 , I4 can be computed analytically
and they have the expressions:
√
√
J1 (θ) = a1 aρ2 + 2bρ + c − ac − ab J.
J2 (θ) =
J3 (θ) =
aρ−3b √
aρ2
2a2
+ 2bρ + c +
3b √
c
2a2
+
3b2 −ac
J.
2a2
2a2 ρ2 − 5abρ + 15b2 − 4ac q 2
15b2 − 4ac √
5b3 − 3bc
aρ
+
2bρ
+
c
−
c
−
J.
6a3
6a3
2a3
where
1
1
aρ + b
b
√ arcsin √
J = √ arcsin √
−
a
a
ac − b2
ac − b2
144
L. Grecu - The boundary element method with linear boundary ...
√
√
b
b + aρ + aρ2 + 2bρ + c
c
1q 2
√
I1 (θ) =
aρ + 2bρ + c −
− √ ln
.
a
a
a a
b + ac
√
I2 (θ) =
c
δ
−√
2
1
bρ + c
ρ2
b
√
+ I2 − I4
,
I
(θ)
=
−
3
2
2
a
aρ + 2bρ + c δ
a aρ + 2bρ + c a
q
√
a (aρ2 + 2bρ + c) + aρ + b
(b2 − δ) ρ + bc
b c
1
√
−
I4 (θ) = √ 2
+ √ ln
(14)
aδ
a a
b + ac
aδ aρ + 2bρ + c
We evaluate the singular integrals. Considering now that the triangle,
noted Tk , has a corner point in xk , which we take as node number one after
a renotation, we calculate the singular integrals occurring in (10) using the
k
following relations: ξ = xk + xj1 − xk r cos θ + xj2 − xk r sin θ , f = f +
f
j2
−f
k
r cos θ + f
j3
−f
k
r sin θ. Denoting by ejk [θ] = xj2 − xk cos θ +
xj3 − xk sin θ, we get rk =
ξ − xk
= krejk (θ)k = r kejk (θ)k, and the
coefficients have in this case the following expressions:
π
Sj
4πν
R2
Bjki =
Sj
4πν
R R
cos θ ρ2 (θ)
dθ,
kejk (θ)k 2
Cjki =
Sj
4πν
R R
sin θ ρ2 (θ)
dθ
kejk (θ)k 2
Ajki
=
0
1
kejk (θ)k
Tj
Tj
qijk = xji 2 − xki
j
Dk
+
Sj
4πν
=
h
Sj
4πν
π
dθ,
R2
[qijk ] sin θ cos θ3 dθ
0 kejk (θ)k
xji 3 − xki
xj3 − xk + xji 2 − xki
xj3 − xk
π
2
i R
0
+
Sj
4πν
h
xji 2
sin2 θ
kejk (θ)k
3
dθ
π
j
Ek
=
Sj
4πν
[qijk ]
R2
sin θ cos2 θ ρ2 (θ)
dθ+
3
2
0 kejk (θ)k
145
−
xj2 − xk .
xki
j3
x −x
k
i Rπ2
cos2 θ
3 dθ
e
0 k jk (θ)k
+
L. Grecu - The boundary element method with linear boundary ...
π
i Z2 cos3 θ ρ2 (θ)
Sj h j2
xi − xki xj3 − xk
+
dθ +
3
4πν
2
ke
(θ)k
jk
0
π
+
Sj
4πν
h
xji 3 − xki
xj3 − x
i Z2 sin2 θ cos θ ρ2 (θ)
k
0
kejk (θ)k3
2
dθ
π
j
Fk
=
Sj
4πν
[qijk ]
R2
sin2 θ cos θ ρ2 (θ)
dθ+
3
2
0 kejk (θ)k
π
+
Sj
4πν
h
xji 2 − xki
xj3 − x
i Z2 cos2 θ sin θ ρ2 (θ)
k
0
π
kejk (θ)k3
2
dθ +
i Z2 sin3 θ ρ2 (θ)
Sj h j3
k
j3
k
xi − xi x − x
dθ
+
3
4πν
2
ke
(θ)k
jk
0
(15)
(16)
As shown in the previous paragraphs all the system’ s coefficients depend
only on the coordinates of the nodes chosen for the boundary discretization,
and on some simple integrals that are one-dimensional and can be evaluated
with a math soft or with the usual rules of the numerical integration. After some manipulation we obtain a linear system of 3M equations with 3
M unknowns,the scalar components of the M nodal values of the unknown
j
functionf , f f1j , f2j , f3j , j = 1, M , having the following form:
M
X
j=1
Aji fij +
M
X
j=1
j
j
B i · f = −νi0 , i = 1, 3,
(17)
where Aji is the coefficient of fij , i = 1, 3 , j = 1, M obtained as a sum of
all the contributions that exist.After solving the system we may compute the
velocity for the M nodes chosen for the discretization of the boundary, and
everywhere on it, and also in the fluid region, and further the pressure on the
boundary and in the fluid region.
146
L. Grecu - The boundary element method with linear boundary ...
References
[1] Brebbia, C.A., Telles, J.C.F., Wobel, L.C.: Boundary Element Theory
and Application in Engineering, Springer-Verlag, Berlin,1984
[2] Grecu, L.: A new boundary element approach for the Stokes flow, Proceedings of ”microCAD 2005” International Scientific Conference, Section G,
63-68
[3] Grecu, L.: A boundary element approach for evaluating the fluid action over the obstacle for the Stokes flow, Proceedings of ”microCAD 2005”
International Scientific Conference, Section G, 69-72
[4] Dragos, L.: Principiile mecanicii mediilor continue, Ed.Tehnica, Bucuresti,1983
[5] Dragos, L.: Metode Matematice in Aerodinamica (Mathematical Methods in Aerodinamics), Editura Academiei Romane, Bucuresti, 2000
[6] Lifanov, I.K.: Singular integral equations and discrete vortices, VSP,
Utrecht, TheNetherlands, 1996
[7] Petrila,T.,Kohr,M.: Some boundary element techniques for Stokes flows
(PartI), Rev. D’Anallyse Numerique et de Theorie de L’Approximation 21(1992),
167-182.
Grecu Luminit¸a
Department of Applied Sciences
University of Craiova, Faculty of Engineering and Management of the Technological Systems, Drobeta Turnu Severin Address: C˘alug˘areni no.1 Dr. Tr.
Severin, cod 220037
email: luminita.grecu@imst.ro; lumigrecu@hotmail.com
147