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