the best estimations of stieltjes functions and teir...
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THE BEST ESTIMATIONS OF STIELTJES FUNCTIONS AND TEIR APPLICATIONS IN MECHANICS OF
INHOMOGENEOUS MEDIA
Institute of Fundamental Technological Research PAS, Warsaw
Stanisław Tokarzewski
1. Subject of investigations
2. Approximation of a Stieltjes function
3. Estimation of a Stieltjes function
4. Relations for inclusion regions
5. Inequalities for Padé approximants
6. Exchangeable power series
7. S- multipoint continued fraction method
8. T- multipoint continued fraction method
9. Comparison with earlier results
10. Conclusion
AbstrtactStarting from the truncated power expansions at real points and infinity it has been established in a unified and coherent form the two methods of estimation of a Stieltjes function called S- and T- multipoint continued fraction methods. As practical applications the bounds on effective transport coefficients of two –phase media with periodical microstructure has been calculated.
1. Subject of investigations
Two-phase compositeTwo-phase composite
.)(Q],,[\z,)u(d,zu
)u(d)z(Q 1110
1
1
0
C
,N,...,,j,x),)xz((O)xz(c)z(Q jp
ji
jij
p
i
jj
211
0
R
pi
i
p
i zO
zc)z(zQ
111
0
.)(Q),z(O)(Q)z(Oc)z(Q )N(i 111111
,
Expansions at Expansions at xxjj
Expansions at infinityExpansions at infinity
Expansion at -1Expansion at -1
Conductivity coefficient
Approximation of conductivity coefficient
1
1
2
z
2. Approximation of Stieltjes functions Two-phase compositeTwo-phase composite
.)(f],,[\z,)u(d,zu
)u(d)z(f
/
1
1
01 0
1C
.N,...,,j,x),)xz((O)xz(c)z(f jp
ji
jij
p
i
jj
211
01
R
.z
Oz
c)z(zfpi
i
p
i
111
01
.)(Q),z(O)(Q)z(Oc)z(f )N(i 111
,
Expansion at Expansion at xxjj
Expansions at infinityExpansions at infinity
Expansion at -1Expansion at -1
Stieltjes functionStieltjes function
Approximation of Stieltjes functions
pp
3. Estimations of Steltjes functions
).z(O)z(f,Oc)z(f
N,...,,j),)xz((O)xz(c)z(f
,)z(f,)z(f,)z(f,...,)z(f,)z(f)z(f
p
z
i
zi
p
i
p
pj
ijij
p
ix
pxxx
,x,...,x,x
,p,...,p,p
jj
jp
j
Np
N
pp
N
N
11
111
01
1
01
111111
1
1
21
2
2
1
1
21
21
.)u(d,)(f,)z(f,zu
)u(d)z(f;f
,,x,...,x
,p,p,...,p
,,x,...,x
p,p,...,p
N
N
N
N
01
11
1
11
011
1
1
1
1
1
).z()z(f,pP),z()z( ,pPj
N
j,,x,...,x,x
p,p,...,p,p
,pP
N
,N
111
1
1 1
21
21
.u);u,z(F)z( ,pP,pP 1111
Notations of trun-cated power expan-sions of f11(z)
Set of all Stieltjes func-tions f11(z) satisfying in-put data
Inclusion region for f11(z)
Complex boundaries for f11(z)
Bounding function for f11(z)111 u);u,z(F ,pP
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
4. Relations for inclusion regions
,
For inclusion regions satisfy the following relations
provided that , ,
Let power expasion of Stieltjes function be given)z(f 1
21
1
2
22
11
1
1
11
1
11
1
11
,,x,...,x
,p,p,...,p,
,,x,...,x
p,p,...,p,
N
IIIIN
II
N
IIN
I
)z(f)z(f,)z(f)z(f
N,...,,j,x],,[\z j 21 RC
),z()z()z(f ,,P
,,P
,III
221111
111
21 21 .PP III
,
-0.1 0.0 0.1 0.2 0.3 0.4
-0.4
-0.3
-0.2
-0.1
-0.0
-0.1-0.8-1.6-2.4-3.2-4.0
-0.1 0.0 0.1 0.2 0.3 0.4
-0.4
-0.3
-0.2
-0.1
-0.0
0.300.280.260.240.220.20
-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3 0.4 0.5 0.6-1.0
-0.8
-0.6
-0.4
-0.2
0.02345
)z(,,P1
.2P,0.3η,0.1ξ4
)u,z(F)g(
z
g)u,z(F
)z(Og(z)f),ξz(Oη(z)f
,,
11
112
111
1
.P,0.3η0.2,2ξ
.)u,z(F
)g(z
g)u,z(F
)z(Og(z)f),ξz(Oη(z)f
,,
2
1 11
112
111
5.P2,0.4694η,1ξ
z
1O
z
14.021
z
1(z)f5),(z0.115(z)f
2),O(z0.187(z)fξ),O(zη(z)f
2
11
11
5. Graphic illustration for fundamental inclusion relations
Padé approksimants and constructed for power series of Stieltjes
satisfy the following inequalities
where step function
6. Basic inequalities for Padé approximants
1)(x,F 1,pP 0)(x,F 1,pP
,,x,...,x
,p,p,...,p
N
IIN
I
)z(f)x(f)x(f
1
1 1
111
),,x(F)()x(f)(),x(F)( ),pP()x(L)x(L
),pP()x(L PPP 11101 111
.pP)xx(Hp)x(L)x(L i
N
ijj
N
jPP 1,1
11
10-1
100
101
102
103
104
105
106
h0
2
4
6
8
10
12
14
16
18
20
Your t ext
1,1,1,1
1,2,1,1
1,2,1,2
1,2,2,2
f
0,2,2,2
0,2,1,2
0,2,1,1
0,1,1,1
depends on input data only
7. Exchangeable power series
,N,...,,j,x),)xz((O)xz(c)z(f jp
ji
jij
p
i
jj
211
01
R
pi
i
p
i zO
zc)z(zf
111
01
.)(Q),z(O)(Q)z(Oc)z(f )N(i 11
,
).)xz
((O)xz
(cz
)z(f p
N
i
Ni
p
i
xN
11
1
01
111111
).)xz((O)xz(c)z(f N!N
N pN
iN)N(i
p
i
x11
111
1
01
2
101111
zO
zcc
z)z(f
,xzOxzx
cxc
x
c)z(f NN
N
N
N
xN 2113
1
110
1
01
1
i
Example
F03,1z,1 c0
1z
1 c1c0
1z
#
F03,1xN1 z,1
c0x N1
1
c0x N1 d1x N1
3z x N1
c0x N1
. #
),C\(z
1,...N)j,min(xξ),)+O((z-x)(z-xc(z)fξxOη(z)f j1p
0ip
ji
jijj j
11 ,)(
Block diagram of the S-multipoint continued fraction method
1
(z)fw
)x(z1
g(z)f PP
j
ip
1Pi
N
1j1 VV
j
1j
Im
Re
Im
Re
Multipoint S-transformations
1
uzFw
)x(z1
guzF P
j
ip
Pi
N
jP VV
j
j
),(),(,
1
111
1
(z)ΦP,1
(z)Φ(z)f 1,PP
(z)Φ1,P
(z)Φ(z)f P,11
Examples3)O(z
7
1(z)fO(z),1(z)f
DataInputInitial
11
)(),(),( zO(z)fzO(z)fzO(z)f
DataInputParametric
111 37
11
(z,u)F3)ξ3η(η
3)1)(z2η(η1
2z1
1(z,u)F
EstimationParametric
1
η,ξ3,1
),()(
,,
uzFz
1
z1
u)(zF
ResultFinal
1
13
332
1
),()(
limlim,,
uzF)(
z)(1
z1
u)(zF
EstimationOptimal
1
013
33312
21
O(z)1(z)f3),O(z7
1(z)f
DataInputInitial
11
)(),(),( zO(z)fzO(z)fzO(z)f
DataInputParametric
111 137
1
),()(,,
zFz1)(zF
ResultFinal
1
13
372
1
7
1
(z,u)F)η1)z2η(η
1
z1
1u)(zF
EstimationOptimal
117
3727
10
13
(
limlim,,
(z,u)F)η1)z2η(η
1
z1
1(z,u)F
EstimationParametric
1
η,ξ3,1
17
3727
1
(
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
-0.0
Re(3,1 (z))
Im(3,1 (z))
Re(3,1 (z))
Im(3,1(z))
x1
x2
x1 x2
3,1(z)
3,1(z)
x1
x2
),C\(z
1,...N)j,min(xξξxOη(z)f
z+O
zc(z)zf),)+O((z-x)(z-xc(z)f
j
1p
0i
pi
i1p
0ip
ji
jij
j
j j
),(
,
1
1111
Block diagram of the T-multipoint continued fraction method
1
(z)fw
)x(z)ex(z1
g(z)f PP
jij
ip
Pi
N
jVV
j
j
11
11
Im
Re
Im
Re
Multipoint T-transformations
1
uzFw
)x(z)ex(z1
guzF P
jij
ip
Pi
N
jpP VV
j
j
),(),(,
1
111
1
(z)Φ ,1pP
(z)Φ(z)f 1,PP
(z)Φ1,P
(z)Φ(z)f ,1pP1
f1z 1z 1 2.5
z ln 12z205z
#
f1z11 0. 4694 Oz 1, f1z2
1 0.18073 Oz 2,
f1z51 0. 11527 Oz 5, f1z2 1
z 1 4. 0236 1z O 1
z 2.
#
F32,1z,u g1
1 z 2e2 z 2g2
1 z 5e3 z 5F1,3z,u,e3
,
F1,3z,u, e3 W3e3F1z 1,u,
g1 0.180733, e2 0. 18073, g2 0.0086, e3 0.0967, W3 0.0111.
#
-0.3 0.0 0.3 0.6
-1.0
-0.7
-0.4
-0.120,1 (z)
21,1 (z)
31,1 (z)
31,1 (z)
f1(z)
Re(P+p ,1(z))
Im(P+p ,1(z))
2+0,1(z)
2+1,1(z)
3+1,1(z)
3+2,1(z)
8
8
z=-3+i
Numerical example
G. Baker, Jr, Essentials of Padé Approximants , ,Academic PressAcademic Press, 1975 , Chapter 17, Section A
G. Baker, Jr, P. Graves-Morris, Padé Approximants , ,CambridgeCambridge PressPress, 1996 , Chapter 5
G. Baker, Jr, Essentials of Padéé Approximants, Approximants, Academic PressAcademic Press, 1975, Chapter 17, Section B
S. Tokarzewski, Continued fraction approach to the bounds on transport coefficients of two phase media,.IFTR Reports 4 (2005)
1
u)(zF
)x(z1
gu)(zF p1
j
iP
1Pi
N
jP VV
1j
,, ,
,
j
11
1
u)(zF
)x(z1
gu)(zF p1
j
iP
1PijP VV
1j
,, ,
,
j
1
11
1
u)(zF
)x(z1
gu)(zF p1
j
ij
ji
N
jP VV
,, ,
,
11
Comparison with the results obtained earlier
1
uzFw
)x(z)ex(z1
guzF P
jij
ip
Pi
N
jpP VV
j
j
),(),(,
1
111
1
S. Tokarzewski, J.J. Telega, S- continued fraction method for the investigation of a complex dielectric constant of two- components composite, Acta Applicandae Mathematicae, 49, 55-83, 1997
S. Tokarzewski, J.J. Telega, M. Pindor, J. Gilewicz, A note on total bounds on complex transport moduli of parametric two-phase media, ZAMP, 54, 713-726, 2003.
0.0 0.5 1.0
Re(1+B0 (z))
0.0
0.5
1.0
Im(1
+B 0 (z
))
z = -1+ia)
0 2 4 6
Re(1+B0 (z))
0
2
4
6
8
10 z = -1+i
b)
0 20 40
Re(1+B0 (z))
0
20
40
60
80
100
z = -1+i
c)
0.04 0.09 0.14 0.19
Re(1+B1 (z))
0.1
0.3
0.5
0.7
0.9
1.1
Im(1
+B 1
(z))
z = -1+i
a)
0 2 4 6
Re(1+B1 (z))
1
3
5
7
9
z = -1+i
b)
0 10 20 30 40 50
Re(1+B1 (z))
0
20
40
60
80
z = -1+i
c)
0.04 0.09 0.14 0.19Re(1+B2 (z))
0.1
0.3
0.5
0.7
0.9
Im(1
+B 2
(z))
z = -1+i
a)
0 2 4Re(1+B2 (z))
1
3
5
7
9
z = -1+i
b)
0 10 20 30 40Re(1+B2 (z))
0
20
40
60
80
z = -1+i
c)Parametric bounds:
02, 0, 0, 0, 1
2, , , , 10
20, 0, 0, 0, 100
A. Gałka, J.J.Telega, S.Tokarzewski, Heat Equation with Temperature-Dependent Conductivity Coefficients and Macroscopic Properties of Microheterogeneous Media, Mathematical and Computer Modelling. 33, 927-942, 1997.
Sequences of Padé approximants forming universal bounds on the effective conductivity of hexagonal array of cylinders with volume fraction =0.88
Telega, J., Tokarzewski, S., and Gałka, A., Modelling torsional properties of human bones by multipoint Padé approximants, In Numerical Analysis and Its Applications (Berlin 2001), L. Vulkov, J. Waśniewski, and P. Yalamov, Eds., Springer-Verlag, pp. 741-748.
Fig.1. Microstructure of a cancellous bone- a,b,c; Three steps Fig.1. Microstructure of a cancellous bone- a,b,c; Three steps of a process of modelling of human bone- d,e,fof a process of modelling of human bone- d,e,f
Effective torsion modulusEffective torsion modulus
Hydraulic stiffening of a model of human bone
Effective torsional complianceEffective torsional compliance
Hydraulic stffening of a model of human bone
S.Tokarzewski, J.J.Telega, Bounds on efffective moduli by analytical continuation of the Stieltjes function expanded at zero and infinity, Z. angew. Math. Phys. 48, 1-20, 1997.
Upper and lower bounds of effective conductivity of square array of cylindersUpper and lower bounds of effective conductivity of square array of cylinders
1.0 2.0 3.0 4.0 5.0 6.0 7.0
1.0
2.0
3.0
4.0
upp. bounds [18/18] 1
asymptotic solution
low. bounds [18/18] 2
=/4=0.785398163397... -touching cylinders
h
=0.78500000
=0.78530000
=0.78539000
=0.78539800
=0.78539816 Q
1. For the first time in literature it is incorporated to estimates of Stieltjes function the
truncated power expansions available at infinity.
2. S- i T- transformation methods are reccurent ones. Hence they do not reqired to solve
the sets of complicated equations.
3. Estimates of Stieltjes functions obtained by means of S- and T- multipoint continued
fraction methods are the best. It means that it is not possible to improve them starting
from given input data.
4. As an examples of applications several numerical computations have been carried out.
The optimal bounds are evaluated on the effective transport coefficients of two phase
media such as dielectric constants of the arrays of spheres, thermal conductivities of
regular lattices of cylinders and rigidities of a porous bars modelling a macroscopical
behaviour of a human bone.
Conclusions