On the generalized method of cells and the prediction of
effective elastic properties of polymer bonded explosives
Biswajit Banerjee 111Corresponding author. E-mail<EMAIL_ADDRESS>Phone: (801)-585-5239 Fax: (801)-585-9826 and Daniel O. Adams
Dept. of Mechanical Engineering, University of Utah,

50 S Central Campus Drive # 2202, Salt Lake City, UT 84112, USA.

## Abstract

The prediction of the effective elastic properties of polymer bonded
explosives using direct numerical simulations is computationally expensive
because of the high volume fraction of particles in these particulate
composites ($\sim$0.90) and the strong modulus contrast between the particles
and the binder ($\sim$20,000). The generalized method of cells (GMC) is an
alternative to direct numerical simulations for the determination of effective
elastic properties of composites. GMC has been shown to be more
computationally efficient than finite element analysis based approaches for a
range of composites. In this investigation, the applicability of GMC to the
determination of effective elastic properties of polymer bonded explosives is
explored. GMC is shown to generate excellent estimates of effective moduli for
composites containing square arrays of disks at volume fractions less than
0.60 and a modulus contrast of approximately 100. However, for high volume
fraction and strong modulus contrast polymer bonded explosives such as PBX
9501, the elastic properties predicted by GMC are found to be considerably
lower than finite element based estimates and experimental data. Simulations
of model microstructures are performed to show that normal stiffnesses are
underestimated by GMC when stress-bridging due to contact between particles is
dominant. Additionally, the computational efficiency of GMC decreases rapidly
with an increase in the number of subcells used to discretize a representative
volume element. The results presented in this work suggest that GMC may not be
suitable for calculating the effective elastic properties of high volume
fraction and strong modulus contrast particulate composites. Finally, a real-
space renormalization group approach called the recursive cell method (RCM) is
explored as an alternative to GMC and shown to provide improved estimates of
the effective properties of models of polymer bonded explosives.

Keywords : Effective Properties, Particulate Composites, High Volume Fraction,
Strong Modulus Contrast, Stress-Bridging, Method of Cells, Real-Space
Renormalization Group

##  1 Introduction

The generalized method of cells (GMC) [Aboudi96_1, Paley92] is a semi-
analytical method of determining the effective properties of composites. In
this method, a representative volume element (RVE) of the composite under
consideration is discretized into a regular grid of subcells. Equilibrium and
compatibility are satisfied on an average basis across subcells using
integrals over subcell boundaries. GMC generates a matrix of algebraic
expressions containing information about subcell material properties. The
effective stiffness of the composite can be obtained by inverting this matrix.

One advantage of GMC over other numerical techniques is that the full set of
effective elastic properties of a composite can be calculated in one step
instead of solving a number of boundary value problems with different boundary
conditions. GMC has also been found to be more computationally efficient that
finite element calculations for fiber reinforced composites [Aboudi96_1,
Wilt95], since far fewer GMC subcells than finite elements are necessary to
obtain the same degree of accuracy. The problem of discretization is also
minimized since a regular rectangular grid is used in GMC.

The generalized method of cells is discussed briefly in this work. Effective
stiffnesses predicted by this method are compared with accurate numerical
predictions for square arrays of disks [Greeng98]. The method is then applied
to two-dimensional models of a general polymer bonded explosive and to the
microstructure of PBX 9501 using a two-step procedure similar to that of Low
et al. [Low94]. GMC estimates of elastic properties are compared with
predictions from detailed finite element calculations. The performance of GMC
is explored for several microstructures with contacting particles and some
shortcomings of the method are identified. Finally, the recursive cell method
(RCM) is explored as an alternative to direct GMC calculations in the
prediction of effective properties of polymer bonded explosives.

##  2 The generalized method of cells

Figure 1 shows a schematic of the RVE, the subcells and the notation
[Aboudi91] used in GMC. In the figure, ($X_{1},X_{2},X_{3}$) is the global
coordinate system of the RVE and
($x^{(\alpha)}_{1}$,$x^{(\beta)}_{2}$,$x^{(\gamma)}_{3}$) is the coordinate
system local to a subcell denoted by $(\alpha\beta\gamma)$. It is assumed that
the displacement function $u^{(\alpha\beta\gamma)}_{i}$ varies linearly within
a subcell $(\alpha\beta\gamma)$ and can be written in the form

$u^{(\alpha\beta\gamma)}_{i}(x^{(\alpha)}_{1},x^{(\beta)}_{2},x^{(\gamma)}_{3})=w^{(\alpha\beta\gamma)}_{i}(X_{1},X_{2},X_{3})+\Phi^{(\alpha\beta\gamma)}_{i}x^{(\alpha)}_{1}+\Theta^{(\alpha\beta\gamma)}_{i}x^{(\beta)}_{2}+\Psi^{(\alpha\beta\gamma)}_{i}x^{(\gamma)}_{3}$
(1)

where $i$ represents the coordinate direction and takes the values 1, 2 or 3;
$w^{(\alpha\beta\gamma)}_{i}$ is the mean displacement at the center of the
subcell $(\alpha\beta\gamma)$; and $\Phi^{(\alpha\beta\gamma)}_{i}$,
$\Theta^{(\alpha\beta\gamma)}_{i}$, and $\Psi^{(\alpha\beta\gamma)}_{i}$ are
constants local to the subcell that represent gradients of displacement across
the subcell. The strain-displacement relations for the subcell are given by

$\epsilon^{(\alpha\beta\gamma)}_{ij}=\frac{1}{2}(\partial_{i}u^{(\alpha\beta\gamma)}_{j}+\partial_{j}u^{(\alpha\beta\gamma)}_{i})$
(2)

where $\partial_{1}=\partial/\partial x^{(\alpha)}_{1}$,
$\partial_{2}=\partial/\partial x^{(\beta)}_{2}$, and
$\partial_{3}=\partial/\partial x^{(\gamma)}_{3}$. Since polymer bonded
explosives are isotropic particulate composites, the following brief
description of GMC assumes that the RVE is cubic and all subcells are of equal
size. If each subcell $(\alpha\beta\gamma)$ has the same dimensions
$(2h,2h,2h)$ then the average strain in the subcell is defined as a volume
average of the strain field over the subcell as

$\left<\epsilon^{(\alpha\beta\gamma)}_{ij}\right>_{\text{S}}=\frac{1}{8h^{3}}\int_{-h}^{h}\int^{h}_{-h}\int_{-h}^{h}{\epsilon^{(\alpha\beta\gamma)}_{ij}}~{}dx^{(\alpha)}_{1}dx^{(\beta)}_{2}dx^{(\gamma)}_{3}$
(3)

The average strain in the subcell can be obtained in terms of the displacement
field variables. It is assumed that there is continuity of traction at the
interface of two subcells. The displacements and tractions are assumed to be
periodic at the boundaries the RVE. Applying the displacement continuity
equations on an average basis over the interfaces between subcells, the
average strain in the RVE can be expressed in terms of the subcell strains.
The average subcell stresses can be obtained from the subcell strains using
the traction continuity condition and the stress-strain relations of the
materials in the subcells. A relationship between the subcell stresses and the
average strains in the RVE is thus obtained.

For orthotropic, transversely isotropic or isotropic materials, the approach
discussed above leads to the decoupling of the normal and shear response of
the RVE. This decoupling leads to two systems of equations relating the
subcell stresses and the average strains in the RVE. For the normal components
of strain, the system of equations can be written as

$\begin{bmatrix}\mathbf{M1}_{1}&\mathbf{M1}_{2}&\mathbf{M1}_{3}\\\
\mathbf{M2}_{1}&\mathbf{M2}_{2}&\mathbf{M2}_{3}\\\
\mathbf{M3}_{1}&\mathbf{M3}_{2}&\mathbf{M3}_{3}\end{bmatrix}\begin{bmatrix}\mathbf{T}_{1}\\\
\mathbf{T}_{2}\\\ \mathbf{T}_{3}\end{bmatrix}=2N\begin{bmatrix}\mathbf{H}\\\
\mathbf{0}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{11}\right>_{\text{V}}+2N\begin{bmatrix}\mathbf{0}\\\
\mathbf{H}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{22}\right>_{\text{V}}+2N\begin{bmatrix}\mathbf{0}\\\
\mathbf{0}\\\ \mathbf{H}\end{bmatrix}\left<\epsilon_{33}\right>_{\text{V}}$
(4)

where $N$ is the number of subcells per side of the RVE. The corresponding
system of equations for the shear components is of the form

$\begin{bmatrix}\mathbf{M4}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{M5}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{M6}\end{bmatrix}\begin{bmatrix}\mathbf{T}_{12}\\\
\mathbf{T}_{23}\\\
\mathbf{T}_{13}\end{bmatrix}=2N^{2}\begin{bmatrix}\mathbf{H}\\\ \mathbf{0}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{12}\right>_{\text{V}}+2N^{2}\begin{bmatrix}\mathbf{0}\\\
\mathbf{H}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{23}\right>_{\text{V}}+2N^{2}\begin{bmatrix}\mathbf{0}\\\
\mathbf{0}\\\ \mathbf{H}\end{bmatrix}\left<\epsilon_{13}\right>_{\text{V}}$
(5)

In equations (4) and (5) the $\mathbf{M}$ matrices contain material compliance
terms. The $\mathbf{T}$ matrices contain the average subcell stresses. The
vector $\mathbf{H}$ contains the dimensions of the subcells. Thus, a sparse
system of equations of size 3$N^{2}$ is produced that relates the subcell
stresses to the average strains in the RVE. After inverting these equations
and with some algebraic manipulation, explicit algebraic expressions for the
individual terms of the effective stiffness matrix can be obtained. These
stress-strain equations that relate the average RVE stresses to the average
RVE strains are of the form

$\begin{bmatrix}\left<\sigma_{11}\right>_{\text{V}}\\\
\left<\sigma_{22}\right>_{\text{V}}\\\ \left<\sigma_{33}\right>_{\text{V}}\\\
\left<\sigma_{23}\right>_{\text{V}}\\\ \left<\sigma_{31}\right>_{\text{V}}\\\
\left<\sigma_{12}\right>_{\text{V}}\end{bmatrix}=\begin{bmatrix}C^{\text{eff}}_{11}&C^{\text{eff}}_{12}&C^{\text{eff}}_{13}&0&0&0\\\
C^{\text{eff}}_{12}&C^{\text{eff}}_{22}&C^{\text{eff}}_{23}&0&0&0\\\
C^{\text{eff}}_{13}&C^{\text{eff}}_{23}&C^{\text{eff}}_{33}&0&0&0\\\
0&0&0&C^{\text{eff}}_{44}&0&0\\\ 0&0&0&0&C^{\text{eff}}_{55}&0\\\
0&0&0&0&0&C^{\text{eff}}_{66}\end{bmatrix}\begin{bmatrix}\left<\epsilon_{11}\right>_{\text{V}}\\\
\left<\epsilon_{22}\right>_{\text{V}}\\\
\left<\epsilon_{33}\right>_{\text{V}}\\\
2\left<\epsilon_{23}\right>_{\text{V}}\\\
2\left<\epsilon_{31}\right>_{\text{V}}\\\
2\left<\epsilon_{12}\right>_{\text{V}}\end{bmatrix}$ (6)

where $C^{\text{eff}}_{ij}$ are the terms of the effective stiffness matrix.
Details of the algebraic expressions for these terms have been published by
other researchers [Pindera97].

In GMC, the number of equations to be solved equals the number of subcells
raised to the $d$ th power, where $d$ is the number of dimensions in the
problem. As a result, the computational efficiency of GMC decreases as the
number of subcells increases. This issue has been partially resolved
[Orozco97] by identifying the sparsity characteristics of the system of
equations and by using the Harwell-Boeing suite of sparse solvers. The
computational efficiency of GMC has been further improved after a
reformulation [Pindera97, Bednar97] that takes advantage of the continuity of
tractions across subcells to obtain a system of $O(N^{2})$ equations in three
dimensions.

Due to decoupling of the normal and shear response of the RVE, the shear
components of the stiffness matrix obtained from GMC are the harmonic means of
the subcell shear stiffnesses and of the form

$1/C^{\text{eff}}_{66}=1/N^{3}\sum^{N}_{\alpha=1}\sum^{N}_{\beta=1}\sum^{N}_{\gamma=1}1/C^{(\alpha\beta\gamma)}_{66}.$
(7)

Bednarcyk and Arnold (2001) suggest that this lack of coupling makes for an
“ultra-efficient” micromechanics model. However, this lack of coupling can
lead to gross underestimation of shear moduli for high volume fraction and
high modulus contrast materials such as polymer bonded explosives. Recently,
researchers [Williams99, Gan00] have attempted to solve the problem by using
higher order expansions for the displacement and by explicitly satisfying both
subcell equilibrium and compatibility. However, these approaches decrease the
computational efficiency of GMC considerably and are not explored in this
work.

##  3 Validation - square arrays of disks

In this section, estimates of effective properties from GMC are compared with
accurate numerical results for square arrays of disks. Square RVEs containing
square arrays of disks exhibit square symmetry. The two-dimensional linear
elastic stress-strain relation for these RVEs can be written as

$\begin{bmatrix}\left<\sigma_{11}\right>_{\text{V}}\\\
\left<\sigma_{22}\right>_{\text{V}}\\\
\left<\sigma_{12}\right>_{\text{V}}\end{bmatrix}=\begin{bmatrix}K_{\text{eff}}+\mu_{\text{eff}}^{(1)}&K_{\text{eff}}-\mu_{\text{eff}}^{(1)}&0\\\
K_{\text{eff}}-\mu_{\text{eff}}^{(1)}&K_{\text{eff}}+\mu_{\text{eff}}^{(1)}&0\\\
0&0&\mu_{\text{eff}}^{(2)}\end{bmatrix}\begin{bmatrix}\left<\epsilon_{11}\right>_{\text{V}}\\\
\left<\epsilon_{22}\right>_{\text{V}}\\\
2\left<\epsilon_{12}\right>_{\text{V}}\end{bmatrix}$ (8)

where $K_{\text{eff}}$ is the two-dimensional effective bulk modulus,
$\mu_{\text{eff}}^{(1)}$ is the effective shear modulus when a shear stress is
applied along the diagonals of the RVE, and $\mu_{\text{eff}}^{(2)}$ is the
effective shear modulus when a shear stress is applied along the edges of the
RVE. These three effective moduli have been determined accurately, using an
integral equation approach, by Greengard and Helsing (1998) for square arrays
of disks containing a range of disk volume fractions.

To compare the effective moduli predicted by GMC with those from the integral
equation calculations [Greeng98], RVEs containing disk volume fractions from
0.10 to 0.70 were created. These RVEs were discretized into 64$\times$64 equal
sized subcells. The effective stiffness matrix of each RVE was calculated
using GMC. Finally, the two-dimensional effective moduli for each RVE were
calculated from the effective stiffness matrix using the relations

$K_{\text{eff}}=0.5(C^{\text{eff}}_{11}+C^{\text{eff}}_{12})~{},~{}\mu^{(1)}_{\text{eff}}=0.5(C^{\text{eff}}_{11}-C^{\text{eff}}_{12})~{},~{}\mu^{(2)}_{\text{eff}}=C^{\text{eff}}_{66}.$
(9)

Figure 2 shows the moduli predicted by GMC and those from the integral
equation method of Greengard and Helsing (G&H) for disk volume fractions from
0.10 to 0.70. The material properties of the disks and the binder used in the
calculations are shown in Table 1. The effective bulk moduli
($K_{\text{eff}}$) and diagonal shear moduli ($\mu_{\text{eff}}^{(1)}$)
obtained from the GMC calculations are within 4% of those obtained by the
integral equation method for all volume fractions up to 0.60. At a volume
fraction of 0.70, the GMC predictions for bulk modulus and diagonal shear
modulus are 4% and 11% less, respectively. For the shear modulus
$\mu_{\text{eff}}^{(2)}$, the GMC predictions are around 4% to 10% less than
the estimates of Greengard and Helsing for volume fractions from 0.10 to 0.60.
The difference is around 24% for a volume fraction of 0.70.

These results show that GMC estimates are quite accurate for composites
containing square arrays of disks with volume fractions up to 0.60, confirming
results reported elsewhere [Aboudi96_1]. In the next section, GMC is used to
determine the effective properties of models of polymer bonded explosives and
the results are compared to detailed finite element calculations and
experimental data.

##  4 Modeling polymer bonded explosives

Polymer bonded explosive (PBX) materials typically contain around 90% by
volume of particles surrounded by a binder. The particles consist of a mixture
of coarse and fine grains with the finer grains forming a filler between
coarser grains. Modeling the microstructure of these materials is difficult
due to the complex shapes of HMX particles and the large range of particle
sizes. Two-dimensional approximations of the microstructure of PBXs based on
digital images [Benson99] have been used to study some aspects of the
micromechanics of PBXs. However, such microstructures are difficult to
generate and require complex image processing techniques and excellent image
quality to accurately capture details of the material. A combination of Monte
Carlo and molecular dynamics techniques have also been used to generate three-
dimensional models of PBXs [Baer01]. Microstructures containing spheres and
oriented cubes have been generated using these techniques and appear to
represent PBX microstructures well. However, the generation of microstructures
using dynamics-based methods is extremely time consuming when tight particle
packing is required, as is the case for volume fractions above 0.70.

Comparisons of finite element predictions with exact relations for the
effective properties of composites [Banerjee02th] have shown that detailed
finite element estimates can be used as a benchmark to check the accuracy of
predictions from GMC. In this investigation, manually generated PBX
microstructures containing symmetrically distributed circular particles are
used initially to compare GMC and finite element predictions. The two-
dimensional microstructures contain 90% particles by volume and use two
particle length scales. Two-dimensional models containing randomly distributed
circular particles that reflect the actual particle size distribution of PBX
9501 are next modeled with GMC and the results compared to finite element
estimates.

The material properties used for the particles, the binder, and PBX9501 in
these calculations are shown in Table 2. These properties correspond to those
of HMX (the explosive particles), the binder, and PBX 9501 at 25o C and a
strain rate of 0.05/s [Wetzel99].

###  4.1 Simplified models of PBX materials

GMC and finite element calculations were performed for the six, manually
generated, simplified model microstructures of polymer bonded explosives shown
in Figure 3. These representative volume elements (RVEs) contain one or a few
relatively large particles surrounded by smaller particles to reflect common
particle size distributions of PBXs. The volume fraction of particles in each
of these models is around 0.90$\pm$0.005. The binder material surrounds all
particles in the six microstructures.

For the GMC calculations, a square grid was overlaid on the RVEs to generate
subcells. Two different approaches were used to assign materials to subcells
before the determination of effective properties of the RVE. In the first
approach, referred to as the “binary subcell approach”, a subcell was assigned
the material properties of particles if more than 50% of the subcell was
occupied by particles. Binder properties were assigned otherwise. Figure 4(a)
shows a schematic of the binary subcell approach. In the second approach,
called the “effective subcell approach”, a method of cells calculation
[Aboudi91] was used to determine the effective properties of a subcell based
on the cumulative volume fraction of particles in the subcell [Banerjee00].
Figure 4(b) shows a schematic of the effective subcell approach. After the
subcells were assigned material properties, the GMC technique was used to
compute the effective properties of the RVE.

Note that the particles are not resolved well when materials are assigned to
subcells in this manner if the number of subcells is small. However, the large
size of the matrix to be inverted in GMC limits the number of subcells that
can be used to discretize the RVE. If the binary subcell approach is used to
assign subcell materials, contacting particles are created where there are
none in the actual microstructure, leading to the prediction of higher than
actual stiffness values. The effective subcell approach improves upon the
binary subcell approach by “smearing”the material properties at the boundaries
of particles and thus reducing the particle contact artifacts caused by
discretization errors.

For validating the GMC results, detailed finite element (FEM) calculations
were performed using six-noded triangular elements to accurately model the
geometry of the particles. Around 65,000 nodes were used to discretize each of
the models. The volume average stress and strain in each RVE was determined
for applied normal and shear displacements. Periodicity was enforced through
displacement boundary conditions. Since these finite element calculations
serve to validate the GMC calculations, further mesh refinement was explored
and the results were found to converge those from the 65,000 node finite
element models.

Table 3 lists the effective stiffnesses of the six RVEs shown in Figure 3 from
GMC and FEM calculations. On average, the GMC calculations using the binary
subcell approach predict values of $C^{\text{eff}}_{11}$ that are around 2.5
times the FEM based values. The values of $C^{\text{eff}}_{11}$ from the
effective subcell approach based GMC calculations are closer to the FEM
estimates than the binary subcell based GMC estimates. The GMC and FEM
estimates of $C^{\text{eff}}_{12}$ are quite close. The values of
$C^{\text{eff}}_{66}$ from GMC are only 10% of the FEM values. The low values
of $C^{\text{eff}}_{66}$ are obtained because GMC predict effective shear
stiffnesses that are harmonic means of subcell shear stiffnesses. Models 3 and
4 (Figure 3) produce agreement in $C^{\text{eff}}_{11}$ and
$C^{\text{eff}}_{12}$ between the binary subcell approach based GMC
calculations and FEM (within 5%) whereas the other four models produce
considerable differences in predictions. These large differences are produced
by discretization errors introduced in the GMC approach that lead to
continuous stress-bridging paths across the RVEs and hence to increased
stiffness.

However, if the predicted effective stiffnesses shown in Table 3 are compared
with the experimental effective stiffness of PBX 9501 (shown in Table 2), it
can be observed that the models predict values of $C^{\text{eff}}_{11}$ and
$C^{\text{eff}}_{12}$ that are around 10% of the experimental values. Hence,
these simplified models are not appropriate for the modeling of PBX 9501. The
next section explores models based on the actual particle size distribution of
PBX 9501.

###  4.2 Models of PBX 9501

Coarse and fine particles of HMX are blended in a ratio of 1:3 by weight and
compacted in the process of manufacturing PBX 9501. Figure 5(a) shows four
RVEs of PBX 9501 based on the particle size distribution of the dry blend of
HMX [Wetzel99] prior to compaction. Figure 5(b) shows four RVEs based on the
particle size distribution of pressed PBX 9501 [Skid98]. The larger particles
are broken up in the pressing process leading to a larger proportion of
smaller particles in pressed PBX 9501. The models of the dry blend have been
labeled “DB” while those of pressed PBX 9501 have been labeled “PP”.

GMC calculations were performed on the PBX 9501 RVEs after discretizing each
RVE into 100$\times$100 subcells and assigning materials to subcells using the
effective subcell approach. In order to validate the GMC predictions, FEM
calculations were also performed on the RVEs after discretizing each RVE into
350$\times$350 four-noded square elements. The binary subcell approach was
applied to assign materials to elements for the FEM calculations.

The particles in each RVE were assigned properties of HMX from Table 2.
However, since particles occupy 92% of the total volume in actual PBX 9501
while the sample microstructures could be filled only up to $\sim$86%, an
intermediate homogenization step was required to determine the properties of
the binder. To produce the desired 92% volume of particles, a fine-particle
filled binder containing 36% particles by volume, or “dirty” binder, was
assumed. The effective elastic properties of the dirty binder were calculated
using the differential effective medium approximation [Markov00].

Table 4 shows the effective stiffness from FEM and GMC calculations for the
models of PBX 9501 shown in Figure 5. For all microstructures, the values of
$C^{\text{eff}}_{11}$ and $C^{\text{eff}}_{22}$ predicted by GMC are less than
5% of the FEM values and less than 10% of the experimental values for PBX 9501
(shown in Table 2). The FEM estimates increase with RVE size (varying from
150% to 450% of the experimental values), reflecting the dependence of
predicted stiffnesses on microstructure, discretization and particle size
distribution. The RVEs contain numerous particle to particle contacts, the
number of which increases with increase in RVE size. These contacts lead to
significant stress-bridging and hence relatively high values of stiffness as
is reflected in the FEM predictions. However, stress-bridging is not
incorporated accurately in the GMC approach leading to considerably lower
values of effective stiffness. The issue of stress-bridging is further
explored in the following section.

The values of $C^{\text{eff}}_{66}$ predicted by GMC are around 0.5% of the
FEM predictions. The reason for this large difference is that the effective
shear stiffness predicted by GMC is simply the harmonic mean of the subcell
shear moduli and only provides a lower bound on the shear stiffness.

##  5 Stress-Bridging

Comparisons of effective stiffness properties predicted by GMC with other
numerical estimates have shown that GMC performs quite well for low modulus
contrast materials with volume fractions below 60%. However, for high modulus
contrast materials with high particle volume fractions, GMC usually predicts
considerably lower effective stiffnesses than finite element calculations. In
this section, GMC is applied to selected microstructures containing stress-
bridging and the predicted properties are compared to finite element
estimates. The goal is to demonstrate that the effects of stress-bridging on
effective properties are inaccurately described by GMC.

###  5.1 Corner bridging : X-shaped microstructure

In the RVE shown in Figure 6, the particles are square, arranged in the form
of an ‘X’, and occupy a volume fraction of 25%. The particles transfer stress
through corner contacts. The effective properties of the X-shaped
microstructure shown in Figure 6 were calculated using the properties of HMX
and five different binders with Young’s moduli that range from 0.7 MPa to 7000
MPa, as shown in Table 5.

Figure 7 shows the variation in the effective stiffness properties
$C^{\text{eff}}_{11}$ and $C^{\text{eff}}_{66}$ of the X-shaped microstructure
with increasing Young’s modulus contrast between the particles and the binder
($E_{p}/E_{b}$). These effective stiffness properties have been calculated
using both finite elements (FEM) (256$\times$256 elements) and GMC
(64$\times$64 subcells). The FEM and GMC estimates are in good agreement for
Young’s modulus contrasts of 200 or less. For higher Young’s modulus
contrasts, the effective stiffness properties predicted by GMC are much lower
than those predicted by FEM. Note that the FEM estimates do not change
significantly with increased discretization, implying that the solution has
converged. The effect of corner singularities is also averaged out while
calculating the effective properties using FEM. If it is assumed that the FEM
estimates are close to the actual effective moduli of the RVE, the GMC
estimates for high modulus contrasts are orders of magnitude lower than the
actual effective moduli. Hence, GMC does not capture the stiffening effect of
corner contacts accurately.

Since the corner stress-bridging problem involves high stress concentrations
that are not resolved well by finite elements, it is possible that the FEM
calculations overestimate the effective properties of the X-shaped
microstructure. Such corner singularities are minimized in the microstructures
studied in the next section where the effect of stress-bridging along
particles edges is studied.

###  5.2 Edge bridging

Figure 8 shows five RVEs (A through E) in which the degree of stress-bridging
is increased progressively from corner bridging, to partial edge bridging, and
finally to continuous stress-bridging across the RVE. In Figure 8, the ‘1’
direction corresponds to the $x$-axis and the ‘2’ direction corresponds to the
$y$-axis.

Model A contains a square particle that occupies 25% of the volume, is
centered in the RVE, and does not have any stress-bridging. Model B contains
three particles that contact along a diagonal of the RVE. In model C, particle
contact is increased to produce a single line of stress-bridging in the
$x$-direction along the center of the RVE. Model D extends the line of contact
in the $x$-direction to an area of contact in the $x$-direction. In Model E,
particle bridging across the RVE is extended to both directions. The material
properties of the constituents of PBX 9501 at room temperature and low strain
rate were used for the calculations (Table 2). GMC simulations of the RVEs
were performed using 100$\times$100 subcells while the validating finite
element calculations were performed using approximately 10,000 eight-noded
quadrilateral elements. Table 6 shows the effective stiffnesses of the five
models obtained from GMC and finite element (FEM) calculations.

As expected in model A, GMC and FEM predict nearly the same values of
effective stiffness since there is no stress-bridging in the model (the
effective stiffness is determined primarily by the volume fraction occupied by
the square particle). However, FEM calculations for model B show that the
diagonal stress-bridge in the model produces a higher stiffness than would
occur if only the volume fraction occupied by the particles were considered in
the calculation of effective stiffness.

The GMC calculations for model B predict values of $C^{\text{eff}}_{11}$ and
$C^{\text{eff}}_{12}$ that are lower than the FEM estimates by a factor of 18.
This discrepancy implies that the diagonal stress-bridge in model B is not
detected by the GMC calculations. The value of $C^{\text{eff}}_{66}$ from FEM
is around 1,400 times that from GMC. This difference shows that, in the
presence of stress-bridging, the shear stiffness can be considerably
underestimated by GMC, even for low volume fraction composites.

Model C has a continuous path through particles along the $x$-axis (the ‘1’
direction) and another continuous particle path along one diagonal.
Intuitively, the stress-bridge path along the ‘1’ direction is expected to
primarily affect the normal components of stiffness ($C^{\text{eff}}_{11}$,
$C^{\text{eff}}_{12}$, $C^{\text{eff}}_{22}$) while particle contact along the
diagonal is expected to affect the shear stiffness ($C^{\text{eff}}_{66}$).
These paths are shown by dashed lines (for normal stress-bridging) and by
dotted lines (for shear stress-bridging) in Figure 9. Results for model C in
Table 6 show that FEM predicts a considerable stiffening in the ‘1’ direction
while GMC does not appear to account for these stress-bridges. Since the shear
stiffness from GMC is simply a harmonic means of the subcell stiffnesses,
$C^{\text{eff}}_{66}$ is not affected at all by geometry and only increases in
proportion with the volume fraction of particles in the RVE.

The estimates of $C^{\text{eff}}_{11}$ for model D and of
$C^{\text{eff}}_{11}$, $C^{\text{eff}}_{12}$, $C^{\text{eff}}_{22}$ for model
E show that GMC can capture the effect of stress-bridging, provided there are
continuous rows of particles with edge-to-edge contacts extending completely
across the RVE.

These studies of stress-bridging explain why GMC underestimates the effective
modulus of the PBX 9501 models shown in Figure 5. In all these models, if
100$\times$100 subcells are used to discretize the RVE, there are no rows or
columns of subcells extending across the RVE that contain no binder. Though
corner contacts and other continuous stress paths exist in the PBX 9501
models, the effects of these stress-bridging paths are not incorporated into
the GMC estimates of effective stiffness. The strain-compatible or shear-
coupled method of cells [Williams99, Gan00] approaches may be able to overcome
some of these deficiencies of GMC. However, the computational efficiency of
GMC is greatly reduced when these modifications are incorporated into GMC and
hence the attractiveness of this micromechanics approach as an alternative to
finite element analysis is also reduced.

##  6 The recursive cell method

The recursive cell method (RCM) [Banerjee02th] is a real-space renormalization
group [Wilson71, Wilson79] approach for calculating the effective elastic
properties of composites that has been developed to address the shortcomings
of GMC while retaining high computational efficiency. A schematic of RCM is
shown in Figure 10. In RCM, as in GMC, the RVE is first discretized into a
regular grid of subcells. For the first iteration of the recursive process,
the subcells are assigned material properties based on the particle
distribution in the RVE using the binary subcell approach discussed earlier.
The subcells in the original grid are then grouped into blocks of n$\times$n
subcells. The effective elastic stiffness matrix of each of the blocks is
calculated using a suitable homogenization approach such as GMC or FEM.
Effective stiffnesses are assigned to each block, resulting in a new, coarser
grid. This procedure is repeated until only one homogeneous block remains. The
properties of this homogeneous block are the effective properties of the RVE.

Studies on the recursive cell method [Banerjee02th] have shown that the method
leads to an upper bound on the effective elastic properties if a FEM approach
is used to homogenize blocks of subcells. As the number of elements used to
discretized a block is increased, the value of the upper bound decreases and a
more accurate estimate of the effective properties is obtained. GMC is an
attractive alternative to the FEM approach for homogenization since less
discretization is required to arrive at the same level of accuracy.

In the previous section, GMC has been shown to not properly account for
stress-bridging in the absence of continuous stress-bridge paths across a RVE.
However, error due to improper stress-bridging is reduced when GMC is used as
the homogenizer in RCM because the probability of the existence of continuous
stress-bridging paths across blocks of subcells is greater than that for the
whole RVE. In addition, homogenization errors due to the overestimation or
underestimation of stress-bridging in sections of the RVE are averaged out if
the particle distribution is sufficiently random.

A second source of error in GMC is the underestimation of the shear stiffness
term $C^{\text{eff}}_{66}$. However, this error can be avoided while using RCM
to determine the effective elastic properties of PBXs because, for
macroscopically isotropic materials such as PBXs, relatively accurate
estimates of the effective shear stiffness can be obtained from the effective
normal stiffness terms [Banerjee02th] and therefore direct estimates of
$C^{\text{eff}}_{66}$ are not required. On the other hand, if the composite is
not macroscopically isotropic, a FEM homogenizer [Banerjee02th] can be used to
determine the value of $C^{\text{eff}}_{66}$ instead of GMC.

The RCM technique has been applied to the four microstructures of the dry
blend and pressed PBX 9501 shown in Figure 5(a) and 5(b), respectively. Each
RVE was discretized into blocks of 256$\times$256 square subcells of equal
size. At each stage of recursion, blocks of 2$\times$2 subcells were
homogenized using GMC.

The values of $C^{\text{eff}}_{11}$ for the four dry blend microstructures
obtained from finite element (FEM) calculations, GMC calculations and RCM
calculations are compared in Figure 11(a). The RCM estimates of
$C^{\text{eff}}_{11}$ for the four microstructures vary from 90% to 150% of
the FEM estimates. These RCM estimates are a considerable improvement over the
GMC estimates shown as black bars in Figure 11(a). Comparisons of
$C^{\text{eff}}_{11}$ for the four pressed PBX 9501 microstructures (shown in
Figure 5(b)) are shown in Figure 11(b). For pressed PBX 9501, the RCM
estimates vary between 84% and 180% of the finite element estimates. These RCM
estimates are also a considerable improvement over the GMC estimates of
effective properties. RCM estimates of $C^{\text{eff}}_{22}$ and
$C^{\text{eff}}_{12}$ for the dry blend and pressed PBX 9501 have also been
found to be in much better agreement with FEM results than the GMC estimates.

As was expected, the estimated value of $C^{\text{eff}}_{66}$ from RCM is
quite low compared to both finite element estimates and experimental data. An
improved estimate of $C^{\text{eff}}_{66}$ can be obtained if the shear
stiffness of each RCM block is calculated using finite elements
[Banerjee02th]. The normal stiffnesses $C^{\text{eff}}_{11}$,
$C^{\text{eff}}_{12}$, and $C^{\text{eff}}_{22}$ can be still be calculated
using GMC, taking advantage of the absence of shear coupling.

These results show that the RCM approach, in conjunction with a GMC
homogenizer, can be used to arrive at reasonably accurate estimates of the
effective properties of PBX materials. The RCM approach can therefore be used
as an alternative to direct GMC calculations for high volume fraction, strong
modulus contrast materials such as polymer bonded explosives.

##  7 Summary and conclusions

The generalized method of cells (GMC) has been found to accurately predict the
effective elastic properties of composites containing square arrays of disks
for volume fractions up to 0.60. However, for two-dimensional models of the
polymer bonded explosive PBX 9501, estimates of effective elastic properties
from GMC have been found to be considerably lower than both experimental
values and estimates based on finite element (FEM) calculations.

The lower values of normal stiffness predicted by GMC for PBX 9501 are due to
inadequate incorporation of particle stress bridging into the approach. Model
representative volume elements (RVEs) with corner and edge stress bridging
show that corner bridging is ignored by GMC while edge stress bridging is
incorporated only if continuous stress bridges exist along entire rows or
columns of subcells that traverse the length of the RVE. Low values of
effective shear stiffness predicted by GMC can be attributed to the use of a
harmonic mean of subcell shear stiffnesses to determine the effective shear
stiffness of a RVE. The harmonic mean is a lower bound on the effective shear
stiffness and is not applicable for microstructures where there is significant
interaction between particles.

Improvements suggested to GMC that incorporate normal-shear coupling and
strain compatibility across subcells have the potential to overcome some of
these weaknesses of GMC. However, these improvements lead to much larger
systems of equations and a considerable increase in the computational cost of
the method. The requirement of inverting a large matrix to obtain the
effective properties makes the generalized method of cells very inefficient as
the number of subcells increases. When materials such as PBX 9501 are modeled,
the number of subcells needed to represent a random distribution of particles
necessarily becomes large. In such situations, the generalized method of cells
becomes inefficient and it may be preferable to perform finite element
analyses to determine the effective properties. Thus, GMC does not appear to
be an improvement over finite element analyses for high volume fraction, high
modulus contrast particulate composites such as polymer bonded explosives.

A computationally efficient alternative to both direct GMC and finite elements
is the recursive cell method (RCM) with GMC being used to homogenize blocks of
subcells. RCM estimates of normal stiffness terms for models of PBX 9501 show
considerable improvement compared to GMC estimates. The RCM estimates of shear
stiffness can be improved if FEM is used, rather than GMC, to determine the
effective shear stiffness of blocks of subcells. RCM, with a combination of
GMC and FEM being used to homogenize blocks of subcells, has the potential of
providing fast and accurate estimates of the effective properties of polymer
bonded explosives.

## Acknlowledgements

This research was supported by the University of Utah Center for the
Simulation of Accidental Fires and Explosions (C-SAFE), funded by Department
of Energy grant DE-FG03-02ER45914. The authors would also like to thank Prof.
Graeme Milton and Dr. Brett Bednarcyk for their suggestions.

## References

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Figure 1: RVE, subcells and notation used in GMC. Figure 2: Comparison of
effective moduli of square arrays of disks from Greengard and Helsing (1998)
(G&H) and GMC calculations. Figure 3: Manually generated microstructures
containing $\sim$ 90% circular particles by volume. Figure 4: Schematics of
the application of the binary subcell approach and the effective subcell
approach in GMC calculations. Figure 5: Microstructures containing circular
particles based on the particle size distribution of the dry blend (DB) of PBX
9501 and of pressed (PP) PBX 9501. Figure 6: RVE used for corner stress-
bridging model. Figure 7: Variation of effective stiffness with modulus
contrast for ‘X’-shaped microstructure. The Young’s modulus contrast is the
ratio of the Young’s modulus of the particles to that of the binder. Figure 8:
Progressive stress-bridging models A through E. Figure 9: Stress-bridging
paths for Model C. Figure 10: Schematic of the recursive cell method.  Figure
11: Comparisons of estimates of $C^{\text{eff}}_{11}$ for (a) models of the
dry blend of PBX 9501 (b) models of pressed PBX 9501.

Table 1: Component properties used by Greengard and Helsing (1998). | Young’s | Poisson’s | Two-Dimensional | Shear
---|---|---|---|---
| Modulus | Ratio | Bulk Modulus | Modulus
| (MPa) |  | (MPa) | (MPa)
Disks | 324 | 0.20 | 225 | 135
Binder | 2.7 | 0.35 | 3.3 | 1
Table 2: Experimentally determined elastic moduli and stiffness of PBX 9501 and its constituents (Wetzel 1999). $C_{ij}$ are components of the stiffness matrix. Material | Young’s | Poisson’s |  $C_{11}$ = $C_{22}$ | $C_{12}$ | $C_{66}$
---|---|---|---|---|---
| Modulus | Ratio |  |  |
| (MPa) |  | (MPa) | (MPa) | (MPa)
Particles | 15300 | 0.32 | 21894 | 10303 | 5795
Binder | 0.7 | 0.49 | 11.97 | 11.51 | 0.235
PBX 9501 | 1013 | 0.35 | 1626 | 875 | 375
Table 3: Effective stiffnesses of the six model microstructures from GMC and FEM calculations. |  $C^{\text{eff}}_{11}$ (MPa) |  $C^{\text{eff}}_{12}$ (MPa) |  $C^{\text{eff}}_{66}$ (MPa)
---|---|---|---
| FEM | GMC | FEM | GMC | FEM | GMC
|  | Binary | Effective |  | Binary | Effective |  | Binary | Effective
|  | Subcell | Subcell |  | Subcell | Subcell |  | Subcell | Subcell
Model 1 | 177 | 814 | 479 | 90 | 119 | 103 | 11 | 2.4 | 2.3
Model 2 | 181 | 807 | 477 | 86 | 112 | 103 | 12 | 2.3 | 2.3
Model 3 | 186 | 815 | 193 | 88 | 108 | 89 | 15 | 2.3 | 2.2
Model 4 | 143 | 116 | 142 | 114 | 112 | 124 | 33 | 2.4 | 2.6
Model 5 | 237 | 132 | 323 | 94 | 100 | 104 | 38 | 2.3 | 2.5
Model 6 | 229 | 132 | 334 | 76 | 93 | 100 | 9 | 2.1 | 2.5
Mean | 192 | 471 | 325 | 91 | 107 | 104 | 20 | 2.3 | 2.4
Table 4: Effective stiffness of the model PBX 9501 microstructures from GMC and FEM calculations. |  $C^{\text{eff}}_{11}$ (MPa) |  $C^{\text{eff}}_{22}$ (MPa) |  $C^{\text{eff}}_{12}$ (MPa) |  $C^{\text{eff}}_{66}$ (MPa)
---|---|---|---|---
| FEM | GMC | FEM | GMC | FEM | GMC | FEM | GMC
Model DB1 | 2385 | 152 | 2094 | 148 | 633 | 122 | 792 | 4.9
Model DB2 | 3618 | 146 | 1643 | 144 | 656 | 122 | 750 | 4.9
Model DB3 | 3546 | 149 | 3385 | 148 | 1142 | 125 | 1317 | 5.0
Model DB4 | 5274 | 144 | 5124 | 146 | 1712 | 120 | 1703 | 4.8
Model PP1 | 3180 | 180 | 3570 | 188 | 989 | 131 | 1262 | 4.8
Model PP2 | 3886 | 170 | 3683 | 190 | 1032 | 132 | 1278 | 5.7
Model PP3 | 6302 | 181 | 6221 | 181 | 2043 | 133 | 2077 | 6.6
Model PP4 | 7347 | 182 | 7587 | 186 | 2547 | 129 | 2542 | 6.9
Table 5: The elastic properties of the components of the X shaped microstructure. $C_{ij}$ are components of the stiffness matrix. | Young’s | Poisson’s | $C_{11}$ | $C_{12}$ | $C_{66}$
---|---|---|---|---|---
| Modulus | Ratio |  |  |
| (MPa) |  | (MPa) | (MPa) | (MPa)
Particles | 15300 | 0.32 | 21894 | 10303 | 5795
Binder a | 0.7 | 0.49 | 12 | 11.5 | 0.2
Binder b | 7 | 0.49 | 120 | 115 | 2.4
Binder c | 70 | 0.49 | 1198 | 1151 | 23.5
Binder d | 700 | 0.49 | 11980 | 11510 | 235
Binder e | 7000 | 0.49 | 119799 | 115101 | 2349
Table 6: Effective properties of edge bridging models. |  $C^{\text{eff}}_{11}$ (MPa) |  $C^{\text{eff}}_{22}$ (MPa) |  $C^{\text{eff}}_{12}$ (MPa) |  $C^{\text{eff}}_{66}$ (MPa)
---|---|---|---|---
| FEM | GMC | FEM | GMC | FEM | GMC | FEM | GMC
Model A | 16 | 16 | 16 | 16 | 15 | 15 | 0.4 | 0.3
Model B | 336 | 19 | 343 | 19 | 337 | 18 | 537 | 0.4
Model C | 4095 | 25 | 889 | 24 | 1470 | 23 | 1093 | 0.5
Model D | 8992 | 8540 | 1361 | 32 | 523 | 23 | 1182 | 0.6
Model E | 10017 | 9042 | 10052 | 9042 | 2892 | 2143 | 1799 | 0.9

Figure 1

Figure 2

Figure 3

(a) Binary subcell approach.

(b) Effective subcell approach.

Figure 4

0.65$\times$0.65 mm2 0.94$\times$0.94 mm2 1.13$\times$1.13 mm2
1.33$\times$1.33 mm2

DB1 DB2 DB3 DB4

(a) Dry Blend of PBX 9501

0.36$\times$0.36 mm2 0.42$\times$0.42 mm2 0.54$\times$0.54 mm2
0.68$\times$0.68 mm2

PP1 PP2 PP3 PP4

(b) Pressed PBX 9501

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

(a) Dry Blend (b) Pressed PBX

Figure 11


# Homotopy properties of the space $OS_{f}(X)$

Kh. Kh. Kurbanov 111Academy of the Armed Forces of Uzbekistan, e-mail:
<EMAIL_ADDRESS>A. Ya. Ishmetov 222Tashkent Institute of Architecture and
Civil Engineering, e-mail<EMAIL_ADDRESS>
###### Abstract

For a given compact Hausdorff space $X$, we construct the space $OS_{f}(X)$ of
normed, order-preserving, weakly additive, positively homogeneous and semi-
additive functionals (for brevity, semi-additive functionals) and it is proved
that the hyperspace $\exp\,X$ of the space $X$ is a deformation retract of the
constructed space. Further we show that the shapes of the spaces $OS_{f}(X)$
and $\exp\,X$ coincide. We establish that if the space $\exp\,X$ is
contractible, then the space $OS_{f}(X)$ is also contractible.

2010 MSC: 54C65, 52A30.

Key words and phrases: semi-additive functional; contractible space; shape.

###### Contents

  1. 1 Introduction
  2. 2 Normality of the functor $OS_{f}$
  3. 3 The contractibility of the space of semiadditive functionals

##  1 Introduction

As it is known, the classical Krein-Milman theorem states that a convex
compact set lying in a locally convex space coincides with the closure of the
convex hull of the set of its extreme points. In the proof of the main
achievements of this paper, this remarkable theorem occupies a central place.
Although this theorem was established for linear spaces, it has recently been
used for wider spaces (see, for example, [3], [5], [8], [11], [12]).

Consider a compact Hausdorff space $X$, a Banach algebra $C(X)$ of all
continuous functions $\varphi\colon\,X\to\mathbb{R}$, provided with pointwise
algebraic operations and a $\sup$-norm, that is, the norm
$\|\varphi\|=\\{|\varphi(X)|:\,x\in X\\}$. For each $c\in\mathbb{R}$, $c_{X}$
denotes the constant function defined by the formula $c_{X}(x)=c$, $x\in X$.
Let $\varphi$, $\psi\in C(X)$. The inequality $\varphi\leq\psi$ means that
$\varphi(x)\leq\psi(x)$ for all $x\in X$.

###### Definition 1.1

A functional $\mu\colon\,C(X)\to\mathbb{R}$ is called:

  1. 1.

weakly additive, if, for all $c\in\mathbb{R}$ and $\varphi\in C(X)$ the
equality $\mu(\varphi+c_{X})=\mu(\varphi)+c$ holds;

  2. 2.

order-preserving  if, for any pair of functions $\varphi$, $\psi\in C(X)$, the
inequality $\varphi\leq\psi$ implies $\mu(\varphi)\leq\mu(\psi)$;

  3. 3.

normed, if $\mu(1_{X})=1$;

  4. 4.

positively homogeneous, if $\mu(t\varphi)=t\mu(\varphi)$ for all $\varphi\in
C(X)$, $t\in\mathbb{R}_{+}$, where $\mathbb{R}_{+}=[0,+\infty)$;

  5. 5.

semiadditive, if $\mu(\varphi+\psi)\leq\mu(\varphi)+\mu(\psi)$ for all
$\varphi$, $\psi\in C(X)$.

For a compact Hausdorff space $X$, by $OS(X)$ we denote the set of all
functionals $\nu\colon\,C(X)\to\mathbb{R}$ that satisfy the above five
conditions, and for brevity, these functionals are called semiadditive
functionals.

The set $OS(X)$ provided with the point-wise convergence topology. Note, the
sets of the view

$\displaystyle\langle\mu;\,\varphi_{1},\,\dots,\,\varphi_{n};\,\varepsilon\rangle=\\{\nu\in
OS(X):\,|\mu(\varphi_{i})-\nu(\varphi_{i})<\varepsilon,\ i=1,\,\dots,\,n\\},$

form a base of neighbourhoods of a functional $\mu\in OS(X)$ in the point-wise
convergence topology, where $\varphi_{i}\in C(X)$, $i=1,\,\dots,\,n$,
$n\in\mathbb{N}$, $\varepsilon>0$.

It is clear, that if $\mu$, $\nu\in OS(X)$, then $\alpha\mu+\beta\nu\in
OS(X)$, where $\alpha$, $\beta\geq 0$, $\alpha+\beta=1$. Moreover, the
following statement is true.

###### Theorem 1.1

[3]. For any compact Hausdorff space $X$, the space $OS(X)$ is a convex
compact with respect to the point-wise convergence topology.

Let $X$ and $Y$ be compact Hausdorff spaces, $f\colon\,X\to Y$ be a continuous
map. By the formula

$\displaystyle OS(f)(\mu)(\varphi)=\mu(\varphi\circ f),\qquad\mu\in OS(X),$

we define a map $OS(f)\colon\,OS(X)\to OS(Y)$, $\varphi\in C(Y)$.

The operation $OS$ defines a covariant functor which acts in the category
$\mathfrak{Comp}$ of compact Hausdorff spaces and their continuous maps. Note,
$OS$ is a normal functor. Obviously, for every compact Hausdorff space $X$,
the space $P(X)$ of probability measures (i. e. linear, non-negative, normed
functionals) is a subspace of $OS(X)$.

Let $A$ be a nonempty subset of the space $P(X)$ of probability measures on
$X$, $\varphi\in C(X)$. Then $|\mu(\varphi)|<\|\varphi\|$ for any $\mu\in A$,
and therefore a number set $\\{\mu(\varphi):\,\mu\in A\\}$ is bounded from
above. Consequently, for every $\varphi\in C(X)$ there is a number

$\displaystyle\nu_{A}(\varphi)=\sup\\{\mu(\varphi):\,\mu\in A\\}.$ (1)

For compact Hausdorff space $X$ by $\exp\,X$ we denote the hyperspace of $X$,
that is, the space of all nonempty closed subsets of $X$ provided by the
Vietoris topology (see [9], [10] for more details). For each $F\in\exp\,X$ we
define a functional $\mu_{F}\colon\,C(X)\to\mathbb{R}$ as following

$\displaystyle\mu_{F}(\varphi)=\max_{x\in F}\varphi(x),\ \ \varphi\in C(X).$
(2)

Clearly, $\mu_{F}$ is a weakly additive, order-preserving, normed, positively-
homogenous and semiadditive functional. The correspondence $F\mapsto\mu_{F}$
is one-to-one. Therefore, we can the set $F$ identify to the functional
$\mu_{F}$. Thus, $\exp\,X\subset OS(X)$.

Let $K$ be a closed subset of some locally convex space $E$. By $cc(K)$ we
denote a set consisting of all convex closed subsets of $K$ and on $cc(K)$
consider the topology induced from $\exp\,K$.

###### Theorem 1.2

[3]. Let $X$ be a compact Hausdorff space. Then spaces $OS(X)$ and $cc(P(X))$
are homeomorphic. This homeomorphism $\tau\colon\,cc(P(X))\to OS(X)$ may by
define by the rule

$\displaystyle\tau(A)=\nu_{A},\qquad\,A\in cc(P(X)).$

From theorem 1.2 and the above mentioned Krein-Milman theorem we get the
following statement.

###### Corollary 1.1

Let $X$ be a compact Hausdorff space. Then spaces $OS(X)$ and $cc(P(X))$ are
homeomorphic. The homeomorphism $\tau\colon\,cc(P(X))\to OS(X)$ may be defined
as

$\displaystyle\tau(A)=\nu_{\operatorname{ext}\,A},\qquad A\in cc(P(X)),$

where $\operatorname{ext}\,A$ is a set of all extreme points of a convex
compact set $A\subset P(X)$.

By reformulating the definition 18 from [15], we introduce the concept of a
support of semiadditive functional. A support of $\mu\in OS(X)$ is a closed
subset $\mbox{supp}\,\mu\subset X$ such that relations
$A\supset\mbox{supp}\,\mu$ and $\mu\in OS(A)$ are equivalent for each closed
$A\subset X$. For a functor the $OS$ the support exists for every $\mu\in
OS(X)$ and it defines as

$\mbox{supp}\,\mu=\cap\left\\{A\subset X:\overline{A}=A,\,\,\mu\in
OS(A)\right\\},$

here $\overline{A}$ is the closure of $A$.

For a compact Hausdorff space $X$ and a positive integer $n$ by $OS_{n}(X)$ we
denote a set of all functionals $\mu\in OS(X)$ for which
$|\operatorname{supp}\,\mu|\leq n$. $OS_{n}(X)$ consider as a subspace of the
space $OS(X)$. Put

$\displaystyle
OS_{\omega}(X)=\underset{i=1}{\overset{\infty}{\cup}}OS_{n}(X).$

A functional $\mu\in OS_{\omega}(X)$ is called as semiadditive functional with
the finite support. Theorem 1.2 and the Corollary 1.1 imply that for each
semiadditive functional $\mu$ with the finite support there exists only unique
convex closed set of $A\in cc(P(X))$ that

$\displaystyle\mu=\nu_{A}=\nu_{\operatorname{ext}\,A}.$

It is clear that

$\displaystyle\operatorname{supp}\,\mu=\underset{\xi\in
A}{\cup}\operatorname{supp}\,\xi.$

At the same time, if each element $\xi\in A$ is a probability measure with the
finite support, assume $\operatorname{supp}\,\xi=\\{x_{\xi
1},\,\dots,\,x_{\xi\,n_{\xi}}\\}$, then

$\displaystyle\xi=\underset{i=1}{\overset{n_{\xi}}{\Sigma}}\alpha_{\xi\,i}\delta_{x_{\xi\,i}},$

where $\underset{i=1}{\overset{n_{\xi}}{\Sigma}}\alpha_{\xi\,i}=1$,
$\alpha_{\xi\,i}\geq 0$, $i=1,\,\dots,\,n_{\xi}$. Hence, the formula (1) can
be written as

$\displaystyle\mu(\varphi)=\sup\left\\{\underset{i=1}{\overset{n_{\xi}}{\Sigma}}\alpha_{\xi\,i}\delta_{x_{\xi\,i}}(\varphi):\,\xi\in
A\right\\}.$ (1${}^{\,\prime}$)

The following set was introduced by E. V. Shchepin.

$P_{f}(X)=\Big{\\{}\mu\in P_{\omega}(X):\,\mbox{ if
}\,\mu=\underset{i=1}{\overset{n}{\Sigma}}\alpha_{i}\delta_{x_{i}},\\\ \mbox{
then there exists }\,i_{0}\in\\{1,\dots,\,n\\}\mbox{ such that
}\,\alpha_{i_{0}}\geq 1-\frac{1}{n+1}\Big{\\}}.$

For a compact Hausdorff space $X$, we define the following set

$\displaystyle OS_{f}(X)=\big{\\{}\nu_{A}\in
OS(X):\,\operatorname{ext}\,A\subset P_{f}(X)\big{\\}}.$

###### Lemma 1.1

For an arbitrary compact Hausdorff space $X$ and every closed $F\subset X$, we
have a $\mu_{F}\in OS_{f}(X)$.

Proof. Let $F\in\exp\,X$. Let us consider the functional $\mu_{F}$ defined by
equality (2). Then $\mu_{F}=\nu_{\\{\delta_{x}:\,x\in F\\}}$. Hence
$\mu_{F}\in OS_{f}(X)$.

$\scriptstyle\blacksquare$

From the proved Lemma 1.1, in particular it follows that
$OS_{f}(X)\neq\varnothing$.

###### Propositon 1.1

The topological space $OS_{f}(X)$, equipped with the point-wise convergence
topology, is a compact Hausdorff space.

Proof. From the homeomorphism $OS_{f}(X)\cong cc(P_{f}(X))$ immediately
follows the proof of the proposition.

$\scriptstyle\blacksquare$

Lemma 1.1 and Proposition 1.1 give the following statement.

###### Corollary 1.2

For an arbitrary compact Hausdorff space $X$, its hyperspace $\exp\,X$ is
homeomorphic to some closed subset of the compact Hausdorff space $OS_{f}(X)$.

Remind the following concept. Let
$\mathfrak{C}=\\{\mathfrak{O},\,\mathfrak{M}\\}$ and
$\mathfrak{C}^{\,\prime}={\\{\mathfrak{O}^{\,\prime},\,\mathfrak{M}^{\,\prime}}\\}$
be two categories, where $\mathfrak{O}$, $\mathfrak{O}^{\,\prime}$ are the
classes of objects, and $\mathfrak{M}$, $\mathfrak{M}^{\,\prime}$ are the
classes of morphisms. A map $F\colon\,\mathfrak{C}\to\mathfrak{C}^{\,\prime}$,
transforming objects to objects, and morphisms to morphisms, is said to be a
covariant functor acting from the category $\mathfrak{C}$ into the category
$\mathfrak{C}^{\,\prime}$ if it satisfies the following conditions:

  * $F1)$

For every morphism $f\colon\,X\to Y$ from the category $\mathfrak{C}$, the
morphism $F(f)$ acts from $F(X)$ to $F(Y)$;

  * $F2)$

$F(\operatorname{id}_{X})=\operatorname{id}_{F(X)}$ for all
$X\in\mathfrak{O}$;

  * $F3)$

$F(f\circ g)=F(f)\circ F(g)$ for every pair of morphisms $f$ and $g$ from
$\mathfrak{M}$.

###### Propositon 1.2

The construction $OS_{f}$ forms a covariant functor acting in the category
$\mathfrak{Comp}$ of compact Hausdorff spaces and their continuous maps.

Proof. At first, for a given map $f\colon\,X\to Y$ we show that
$OS(f)(OS_{f}(X))\subset OS_{f}(Y)$. Let $\mu\in OS_{f}(X)$. Then there exists
an $A\subset P(X)$ such that $\operatorname{ext}\,A\subset P_{f}(X)$ and
$\mu=\nu_{\operatorname{ext}\,A}$. But, then from the results of work [7] it
follows that $P_{f}(f)(\operatorname{ext}\,A)=\operatorname{ext}\,f(A)\subset
P_{f}(Y)$. Therefore,
$OS_{f}(\mu)=OS_{f}(\nu_{\operatorname{ext}\,A})=\nu_{\operatorname{ext}\,f(A)}\in
OS_{f}(Y)$. Now let us define a map $OS_{f}(f)\colon\,OS_{f}(X)\to OS_{f}(Y)$
as the restriction $OS_{f}(f)=OS(f)|_{OS_{f}(X)}$. Since $OS(f)$ is continuous
[3], its restriction $OS_{f}(f)$ is also continuous. Thus, $OS_{f}$ satisfies
the condition $F1)$.

Let $\operatorname{id}_{X}\colon\,X\to X$ be the identity map. For every
$\mu\in OS_{f}(X)$ we have

$\displaystyle
OS_{f}(\operatorname{id}_{X})(\mu)(\varphi)=\mu(\varphi\circ\operatorname{id}_{X})=\mu(\varphi),\qquad\varphi\in
C(X).$

Since $\mu$ and $\varphi$ are arbitrary, then it becomes
$OS_{f}(\operatorname{id}_{X})(\mu)=\mu$ (the condition $F2)$ carried out).

Let us show that $OS_{f}$ preserves the map composition. Let $X$, $Y$, $Z$ be
compact Hausdorff spaces and $f\colon\,X\to Y$, $g\colon\,Y\to Z$ be
continuous maps. For $\mu\in OS_{f}(X)$ and $\varphi\in C(Z)$ we have

$OS_{f}(g\circ f)(\mu)(\varphi)=\mu(\varphi\circ(g\circ f))=\mu((\varphi\circ
g)\circ f)=OS_{f}(f)(\mu)(\varphi\circ g)=\\\ =OS_{f}(g)\circ
OS_{f}(f)(\mu)(\varphi),$

i. e. $OS_{f}(g\circ f)=OS_{f}(g)\circ OS_{f}(f)$ (the condition $F3)$ is
established).

$\scriptstyle\blacksquare$

Thus, the construction $OS_{f}$ transforming compact Hausdorff spaces to
compact Hausdorff spaces, and continuous maps of compact Hausdorff spaces in
continuous maps of compact Hausdorff spaces, forms a covariant functor acting
in the category of compact Hausdorff spaces and their continuous mappings.

In the works of [15] E. V. Schenpin, [13] V. V. Fedorchuk, [7] A. A. Zaitov,
[4] T. F. Jurayev were studied the functor $P_{f}$ a traditional probability
analogue of the built functor $OS_{f}$. The work of [6] A. A. Zaitov and A.
Ya. Ishmetov was devoted to the idempotent analogue $I_{f}$.

In the present work, we establish that the functor $OS_{f}$ is a normal in the
category of compact Hausdorff spaces and their continuous maps. Further we
prove that the hyperspace $\exp\,X$ of the compact Hausdorff space $X$ is a
deformation retract of the space $OS_{f}(X)$. We also show that the shapes of
$OS_{f}(X)$ and $\exp\,X$ are the same. We prove that if $\exp\,X$ is
contractible, then $OS_{f}(X)$ is also contractible. Obtained results strictly
differ from the above-described authors’ results, because in previous works
have been established relationships between the compact Hausdorff space $X$
and $I_{f}(X)$ or $P_{f}(X)$.

But all these three functors have one remarkable property: the degrees of the
functors $OS_{f}$, $P_{f}$ and $I_{f}$ are infinite. But this phenomenon
immediately generates another difference: the functors $P_{f}$ and $I_{f}$ are
with a finite support, and the construction $OS_{f}$ is a functor with an
infinite support.

Recall the concept of the degree of functors. Let $X$ be a compact Hausdorff
space, $F$ is a functor and $x\in F(X)$. A degree of the point $x$ is a
smallest positive integer $n$ such that $x$ belongs to $F(f)F(K)$ for some map
$f\colon K\to X$ of $n$-point space in $K$ ([15], Definition 16). If does not
exist such finite $n$ , then degree of $x$ is considered as infinite. Degree
of a functor of $F$ is the maximum of degrees of various points $x\in F(X)$
for various compact Hausdorff spaces $X$ and it denotes by $\mbox{deg}F$.

##  2 Normality of the functor $OS_{f}$

A functor $F\colon\mathfrak{Comp}\to\mathfrak{Comp}$ acting in the category of
compact Hausdorff spaces and their continuous maps is said to be normal if it
satisfies the following conditions ([15], Definition 14):

  1. 1.

$F$ is continuous ($F(\lim S)=\lim F(S)$),

  2. 2.

$F$ preserves weight ($wX=wF(X)$),

  3. 3.

$F$ is monomorphic (i. e. preserves the injectivity of maps),

  4. 4.

$F$ is epimorphic (i. e. preserves the surjectivity of maps),

  5. 5.

$F$ preserves the intersections
($F(\bigcap\limits_{\alpha}X_{\alpha})=\bigcap\limits_{\alpha}(F(X_{\alpha})$),

  6. 6.

$F$ preserves the preimages ($F(f^{-1})=F(f)^{-1}$),

  7. 7.

$F$ preserves a point and an empty set ($F(\mathbf{1})=\mathbf{1}$,
$F(\varnothing)=\varnothing$).

Let us decipher this definition. Let
$S=\\{X_{\alpha},\,p_{\alpha}^{\beta};\,\mathfrak{A}\\}$ be the inverse
spectrum of compact Hausdorff spaces, $\lim S=\lim\limits_{\leftarrow}S$ is
its limit. According to Kurosh theorem, the inverse spectrum limit of non-
empty compact Hausdorff spaces is non-empty ([14], Theorem 3.13) and is a
compact Hausdorff space ([14], Proposition 3.12). Under the impact of the
functor $F$ to the compact Hausdorff spaces $X_{\alpha}$ and to the maps
$p_{\alpha}^{\beta}$, $\alpha,\,\beta\in\mathfrak{A}$, $\alpha\prec\beta$, the
inverse spectrum
$F(S)=\\{F(X_{\alpha}),\,F(p_{\alpha}^{\beta});\,\mathfrak{A}\\}$ is formed.
Let $\lim F(S)$ be the limit of this spectrum. Condition 1 requires that the
equality $F(\lim S)=\lim F(S)$ has to hold. For a topological space $X$ by
$wX$ we denote its weight, that is, the smallest of the base powers of the
space $X$. Condition 2 requires that the weights of the compact Hausdorff
spaces $X$ and $F(X)$ be equal. The monomorphism of the functor $F$ (condition
3) allows us to consider $F(A)$ as the subspace $F(X)$ for the closed
$A\subset X$. The identity of $F(A)$ with the subspace of $F(X)$ is done by
the idendity $F(i_{A})$, where $i_{A}\colon A\to X$ is the identity. Condition
4 requires that if $f\colon X\to Y$ is a continuous map “onto”, then
$F(f)\colon F(X)\to F(Y)$ is also a continuous map “onto”. For a monomorphic
functor $F$, conditions 5 and 6 are deciphered as follows: for any family
$\\{X_{\alpha}\\}$ of closed subsets of an arbitrary compact Hausdorff space
$X$, the equality
$F(\bigcap\limits_{\alpha}X_{\alpha})=\bigcap\limits_{\alpha}F(X_{\alpha})$
has to hold (condition 5); For every continuous map of $f\colon X\to Y$ and
every closed $B$ in $Y$, the equality $F(f^{-1}(B))=F(f)^{-1}F(B)$ (condition
6) is true. The point preservation condition means that $F$ takes a one-point
space to a one-point space.

###### Propositon 2.1

The functor $OS_{f}$ preserves the weight of infinite compact Hausdorff
spaces, i. e. for every infinite compact Hausdorff space the equality
$w(OS_{f}(X))=w(X)$ holds.

Proof. From the relations $X\cong\delta(X)\subset OS_{f}(X)\subset OS(X)$ and
the equality $w(OS(X))=w(X)$ established in [3] follows the required equality.

$\scriptstyle\blacksquare$

###### Propositon 2.2

$OS_{f}$ is a monomorphic functor, i. e., it preserves the injectivity of
mappings of compact Hausdorff spaces.

Proof. Let $\mu_{1}$, $\mu_{2}\in OS_{f}(X)$, $\mu_{1}\neq\mu_{2}$. Owing to
the injectivity of the map $f$ there exists a function $\varphi\in C(Y)$, such
that $\mu_{1}(\varphi\circ f)\neq\mu_{2}(\varphi\circ f)$. Hence
$OS_{f}(f)(\mu_{1})(\varphi)=\mu_{1}(\varphi\circ f)\neq\mu_{2}(\varphi\circ
f)=OS_{f}(f)(\mu_{2})(\varphi)$.

$\scriptstyle\blacksquare$

###### Propositon 2.3

If $f\colon\,X\to Y$ is a continuous map “onto”, then
$OS_{f}(f)\colon\,OS_{f}(X)\to OS_{f}(Y)$ is also a continuous “onto” map.

Proof. The continuity of the map $OS_{f}(f)$ had shown in Proposition 1.2.
Since the for a surjective map $f\colon\,X\to Y$ the map $OS(f)$ is surjective
[3], then its restriction $OS_{f}(f)$ is also surjective.

$\scriptstyle\blacksquare$

###### Propositon 2.4

Functor $OS_{f}\colon\,\mathfrak{Comp}\to\mathfrak{Comp}$ preserves

  * $a)$

a point,

  * $b)$

the empty set.

Proof. a) Let $x\in X$. By definition, we have
$OS_{f}(\\{x\\})=\\{\delta_{x}\\}$.

b) Let $X=\varnothing$. Then $C(X)=\varnothing$. Consequently,
$\mathbb{R}^{C(X)}=\mathbb{R}^{\varnothing}=\varnothing$. From here we get
$OS_{f}(\varnothing)\subset\varnothing$.

$\scriptstyle\blacksquare$

###### Propositon 2.5

If $A$ is a closed subset of a compact Hausdorff space $X$, then
$OS_{f}(A)\subset OS_{f}(X)$.

Proof. Let $A$ be closed in $X$ and $\mu\in OS_{f}(A)$. Then the functional
$\mu$ is concentrated on $A$. Owing to the definition of the concept of the
support it is equivalent to $\operatorname{supp}\,\mu\subset A$. Then
$\operatorname{supp}\,\mu\subset X$, from where $\mu\in OS_{f}(X)$.

$\scriptstyle\blacksquare$

###### Propositon 2.6

If $f\colon\,X\to Y$ is a continuous map between compact Hausdorff spaces and
$B\subset Y$, then $OS_{f}(f^{-1}(B))=OS_{f}(f)^{-1}(OS_{f}(B))$.

Proof. Let $\mu\in OS_{f}(f^{-1}(B))$. By the definition, this means that
$\mu\in OS_{f}(X)$ and $\operatorname{supp}\,\mu\subset f^{-1}(B)$.
Consequently, $f(\operatorname{supp}\,\mu)\subset B$. Therefore
$\operatorname{supp}\,OS_{f}(f)(\mu)\subset B$. From here $OS_{f}(f)(\mu)\in
OS_{f}(B)$, i. e. $\mu\in OS_{f}(f)^{-1}(OS_{f}(B))$.

Inversely, let $\mu\in OS_{f}(f)^{-1}(OS_{f}(B))$. Then $OS_{f}(f)(\mu)\in
OS_{f}(B)$, i. e. $\operatorname{supp}\,OS_{f}(f)(\mu)\subset B$.
Consequently, $f(\operatorname{supp}\,\mu)\subset B$. This means that
$\operatorname{supp}\,\mu\subset f^{-1}(B)$, whence $\mu\in
OS_{f}(f^{-1}(B))$.

$\scriptstyle\blacksquare$

Let $\\{X_{\alpha},\,p_{\alpha}^{\beta};\,A\\}$ be an inverse spectrum indexed
by the elements of the set $A$ and consisting of compact Hausdorff spaces. By
$\lim X_{\alpha}$ we denote the limit of this spectrum, and by
$p_{\alpha}\colon\,\lim X_{\alpha}\to X_{\alpha},\alpha\in A$, the limit
projections. Inverse spectrum $\\{X_{\alpha},\,p_{\alpha}^{\beta};\,A\\}$
generates the inverse spectrum
$\\{OS_{f}(X_{\alpha}),\,OS_{f}(p_{\alpha}^{\beta});\,A\\}$, which limit is
denoted by $\lim OS_{f}(X_{\alpha})$, and the limit projections by
$pr_{\alpha}\colon\,\lim OS_{f}(X_{\alpha})\to OS_{f}(X_{\alpha})$. The maps
$OS_{f}(p_{\alpha})\colon\,OS_{f}(\lim X_{\alpha})\to
OS_{f}(X_{\alpha}),\colon\,\alpha\in A$, generate the map
$R_{OS_{f}}\colon\,OS_{f}(\lim X_{\alpha})\to\lim OS_{f}(X_{\alpha})$.

###### Propositon 2.7

The functor $OS_{f}$ is continuous, i. e., the map
$R_{OS_{f}}\colon\,OS_{f}(\lim X_{\alpha})\to\lim OS_{f}(X_{\alpha})$ is a
homeomorphism.

Proof. Since taking an affine combination and taking a closure are continuous
operations, it follows from the continuity [3] of the functor $OS$ that
$R_{OS_{f}}\colon\,OS_{f}(\lim X_{\alpha})\to\lim OS_{f}(X_{\alpha})$ is a
homeomorphism.

$\scriptstyle\blacksquare$

###### Propositon 2.8

The functor $OS_{f}$ preserves the intersection, i. e., for any pair of closed
subsets $A$, $B$ of a compact Hausdorff space $X$, we have

$\displaystyle OS_{f}(A\cap B)=OS_{f}(A)\cap OS_{f}(B).$

Proof. The inclusion $OS_{f}(A\cap B)\subset OS_{f}(A)\cap OS_{f}(B)$ is
clear. If $\mu\in OS_{f}(A)\cap OS_{f}(B)$, then
$\operatorname{supp}\,\mu\subset A$ and $\operatorname{supp}\,\mu\subset B$;
consequently, $\operatorname{supp}\,\mu\subset A\cap B$. From here $\mu\in
OS_{f}(A\cap B)$, i. e. $OS_{f}(A\cap B)\supset OS_{f}(A)\cap OS_{f}(B)$.

$\scriptstyle\blacksquare$

Thus, the following main result of the section is proved.

###### Theorem 2.1

$OS_{f}\colon\,\mathfrak{Comp}\to\mathfrak{Comp}$ is a normal functor.

##  3 The contractibility of the space of semiadditive functionals

In this section, we will establish that if for a given Hausdorff compact space
$X$ its hyperspace $\exp\,X$ is a contractible compact, then $OS_{f}(X)$ is
also a contractible compact.

A subset $Y$ of a topological space $X$ is ([2], p. 14) a retract of $X$ if
there exists a map $r\colon X\to Y$ (called a retraction of $X$ into $Y$) such
that the restriction $r|_{Y}\colon Y\to Y$ is the identity map $id_{Y}\colon
Y\to Y$ (i. e., $r(y)=y$ for all $y\in Y$). If $r\colon X\to Y$ is a
retraction and there exists a homotopy $h\colon X\times[0,\,1]\to Y$ such that
$h(x,0)=x$, $h(x,1)=r(x)$, for all $x\in X$, then $r$ is a deformation
retraction, and $Y$ is a deformation retract of the space $X$. A deformation
retraction $r\colon X\to F$ is a strongly deformation retraction if, for the
homotopy $h\colon X\times[0,\,1]\to Y$, we have $h(x,t)=x$ for all $x\in F$
and all $t\in[0,\,1]$. A space $Y$ is an absolute retract (and they write
$Y\in AR$) if, for every homeomorphism $h$ that maps $Y$ onto a closed subset
$hY$ of any space $X$, the set $hY$ is a retract of the space $X$. A space $Y$
is called an absolute neighborhood retract (and they write $Y\in ANR$) if, for
every homeomorphism $h$ mapping $Y$ onto a closed subset $hY$ of any space
$X$, there exists a neighborhood $U$ of the set $hY$ (in $X$) such that $hY$
is a retract for $U$.

Let $X$ and $Y$ be two compact sets lying in the metrizable spaces $M$ and
$N$, respectively, where $M,\,N\in AR$. The sequence of mappings $f_{k}\colon
M\to N$, $k=1,\,2,\,\dots,$ is called ([2], p. 17) the fundamental sequence
from $X$ to $Y$, if for each neighborhood $V$ of the compactum $Y$ (in $N$)
there exists a neighborhood $U$ of the compactum $X$ (in $M$) such that

$f_{k}|_{U}\simeq f_{k+1}|_{U}\quad\mbox{in}\quad V\quad\mbox{for almost
all}\quad k=1,\,2,\,\dots.$

Here, “for almost all” means “for all but finite number”. The relation
$f_{k}|_{U}\simeq f_{k+1}|_{U}$ means that there exists a homotopy
$\varphi_{k}\colon U\times[0,\,1]\to V$ such that
$\varphi_{k}(x,\,0)=f_{k}(x)$ and $\varphi_{k}(x,\,1)=f_{k+1}(x)$ for all
$x\in U$. This fundamental sequence is denoted by
$\left\\{f_{k},\,X,\,Y\right\\}_{M,\,N}$ or shortly by $\mathbf{f}$, and they
write $\mathbf{f}\colon X\to Y$ in $M$, $N$. They say that the fundamental
sequence $\mathbf{f}=\left\\{f_{k},\,X,\,Y\right\\}_{M,\,N}$ is generated by
the map $f\colon X\to Y$ if $f_{k}(x)=f(x)$ for all $x\in X$ and for all
$k=1,\,2,\,\dots$.

Let $X$ and $Y$ be closed subsets of metrizable $AR$-spaces $M$ and $N$,
respectively. They say ([2], p. 29) that the spaces $X$ and $Y$ are
fundamental equivalent (with respect to $M$, $N$) if there exist two
fundamental sequences $\mathbf{f}\colon X\to Y$ and $\mathbf{g}\colon Y\to X$
such that $\mathbf{g}\mathbf{f}=id_{X,\,M}$ and
$\mathbf{f}\mathbf{g}=id_{Y,\,N}$. The fundamental equivalence relation is
equivalence relation. Therefore, the class of all spaces splits into pairwise
disjoint classes of spaces, which are called shapes ([2], p. 31).
Consequently, two spaces belong to the same shape if and only if they are
fundamental equivalent. The shape containing the space $X$ is called the shape
of the space $X$ and it denotes by $Sh(X)$. The concept of shape is
topological, i. e., two homeomorphic spaces have the same shape. It is known
that for two absolute neighborhood retracts $A$ and $B$, the equality
$Sh(A)=Sh(B)$ holds if and only if they are homotopy equivalent.

Take an arbitrary functional $\nu_{A}\in OS_{f}(X)$. Note that each
probability measure $\xi$ with finite support, say,
$\\{x_{1},\,\dots,\,x_{n}\\}$, is represented in the form of an affine
combination $\xi=\overset{n}{\underset{i=1}{\Sigma}}\alpha_{i}\delta_{x_{i}}$
of Dirac measures $\delta_{x_{i}}$, $i=1,\,\dots,\,n$, uniquely ( see, for
example, [13]).

Let $\operatorname{ext}\,A=\\{\xi_{s}:\,s\in S\\}$. By definition,
$\operatorname{ext}\,A\subset P_{f}(X)$. It is clear that the support of each
$\xi_{s}\in\operatorname{ext}\,A$ is finite; suppose
$|\operatorname{supp}\,\xi_{s}|=n_{s}$. Let
$\xi_{s}=\overset{n_{s}}{\underset{i=1}{\Sigma}}\alpha_{s,i}\delta_{x_{s,i}}$,
$s\in S$. By construction, for each $\xi_{s}$ there exists
$i(s)\in\\{1,\,\dots,\,n_{s}\\}$ such that $\alpha_{k,i(s)}\geq
1-\frac{1}{n_{s}+1}$. To the functional $\nu_{A}$ we associate the set
$F_{A}:=\\{x_{1,i(s)}:\,s\in S\\}$. The defined correspondence
$OS_{f}(X)\to\exp\,X$ is denoted by $r_{e}^{O}=r_{\exp\,X}^{OS_{f}(X)}$. The
map $r_{e}^{O}\colon OS_{f}(X)\to\exp\,X$ is defined correctly. The
construction of the map $r_{e}^{O}$, easily implies that
$r_{e}^{O}(\mu_{F})=F$ for each $F\in\exp\,X$, i. e., under the map
$r_{e}^{O}$, the points of the space $\exp\,X$ are fixed points, where
$\mu_{F}$ is the functional defined by equality (2). Consequently, $r_{e}^{O}$
is a retraction, and the set $\exp\,X$ is a retract of the set $OS_{f}(X)$.

We will establish a stronger statement. To prove it, we identify the set
$F\in\exp\,X$ with the functional $\mu_{F}\in OS_{f}(X)$.

###### Theorem 3.1

For an arbitrary Hausdorff compact space $X$, the set $\exp\,X$ is a strongly
deformation retract of the Hausdorff compact space $OS_{f}(X)$.

Proof. Consider a map $h\colon OS_{f}(X)\times[0,\,1]\to OS_{f}(X)$ defined by
the formula

$\displaystyle h(\mu,\,t)=h_{t}(\mu)=(1-t)\cdot\mu+t\cdot
r_{e}^{O}(\mu),\qquad(\mu,\,t)\in OS_{f}(X)\times[0,\,1].$

It is easy to verify that the map $h$ is well defined. Moreover,
$h_{0}=\operatorname{id}_{OS_{f}(X)}$ and $h_{1}=r_{e}^{O}$, i. e., $h$ is the
homotopy connecting maps $\operatorname{id}_{OS_{f}(X)}$ and $r_{e}^{O}$.
Further, we have

$h(\mu_{F},\,t)=(1-t)\cdot\mu_{F}+t\cdot r_{e}^{O}(\mu_{F})=\mu_{F},$

i. e., $h_{t}(\mu_{F})=\mu_{F}$ for all $F\in\exp\,X$ and $t\in[0,\,1]$. Thus,
$\exp\,X$ is a strongly deformation retract of the compact $OS_{f}(X)$.

$\scriptstyle\blacksquare$

From Theorem 3.1 and statement (5.4) from [2] (p. 32) we obtain

###### Corollary 3.1

For an arbitrary compact Hausdorff space $X$, we have

$Sh(\exp\,X)=Sh(OS_{f}(X)).$

Recall ([1], p. 29) that a $A\subset X$ is contractible in the space $X$ to to
the set $B\subset X$ if the embedding $i_{A}\colon A\to X$ is homotopic to
some map $f\colon A\to X$ such that $f(A)\subset B$. If in this case $B$
consists of only one point, then they say that $A$ is contractible in $X$.

Clearly, if there exists a homotopy $h\colon A\times[0;\,1]\to A$, such that
$h(y,0)=i_{A}$, and $h(y,1)=\\{\text{point}\\}$, then $A$ is contractible in
$X$.

A space $X$ is called ([1], p. 31) locally contractible at a point $x_{0}\in
X$ if every neighborhood $U$ of the point $x_{0}$ contains a neighborhood
$U_{0}$ contractible in $U$ to a point. A space $X$ is called locally
contractible if it is locally contractible at each of its points.

###### Theorem 3.2

If for a Hausdorff compact space $X$ its hyperspace $\exp\,X$ is contractible,
then the space $OS_{f}(X)$ is also contractible.

Proof. We show more: the functor $OS_{f}$ preserves the homotopy of maps. Let
$h_{0}$, $h_{1}\colon X\to Y$ be homotopical maps, $h\colon X\times[0,\,1]\to
Y$ be the homotopy connecting the maps $h_{0},\ \ h_{1}$, i. e.
$h(x,\,0)=h_{0}(x)$, $h(x,\,1)=h_{1}(x)$. The embedding $i_{t_{0}}\colon
X\times\\{t_{0}\\}\to X\times I$ defined by the equality
$i_{t_{0}}(x,\,t_{0})=(x,\,t_{0})$, $x\in X$, defines the embedding
$OS_{f}(i_{t_{0}})\colon OS_{f}(X\times\\{t_{0}\\})\to OS_{f}(X\times I)$.
But, for every $t_{0}\in[0,\,1]$, the space $OS_{f}(X\times\\{t_{0}\\})$ is
naturally homeomorphic to $OS_{f}(X)\times\\{t_{0}\\}$. This homeomorphism can
be realized, as it is easy to see, using equality (1${}^{\,\prime}$) and the
correspondence $\mu_{t_{0}}\leftrightarrow(\mu,\,t_{0})$, where for each
$\\{\xi_{s}:\,s\in S\\}\subset P_{f}(X)$:

$\displaystyle\mu(\varphi)$
$\displaystyle=\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{x_{s\,i}}(\varphi):\,s\in
S\right\\},\qquad\varphi\in C(X),$ $\displaystyle\mu_{t_{0}}(\phi)$
$\displaystyle=\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{(x_{s\,i},\,t_{0})}(\phi):\,s\in
S\right\\},\qquad\phi\in C(X\times\\{t_{0}\\}).$

We now define a map $OS_{f}(h)\colon OS_{f}(X)\times[0,\,1]\to OS_{f}(Y)$ by
the equality

$OS_{f}(h)\left(\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{x_{s\,i}}:\,s\in
S\right\\},\,t\right)=\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{h(x_{s\,i},\,t)}:\,s\in
S\right\\}.$

We have

$\displaystyle
OS_{f}(h)\left(\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{x_{s\,i}}:\,s\in
S\right\\},\,0\right)=\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{h(x_{s\,i},\,0)}:\,s\in
S\right\\}=$
$\displaystyle=\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{h_{0}(x_{s\,i})}:\,s\in
S\right\\}=OS_{f}(h_{0})\left(\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{x_{s\,i}}:\,s\in
S\right\\}\right),$ $\displaystyle
OS_{f}(h)\left(\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{x_{s\,i}}:\,s\in
S\right\\},\,1\right)=\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{h(x_{s\,i},\,1)}:\,s\in
S\right\\}=$
$\displaystyle=\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{h_{1}(x_{s\,i})}:\,s\in
S\right\\}=OS_{f}(h_{1})\left(\sup\left\\{\underset{i=1}{\overset{n_{s}}{\Sigma}}\alpha_{s\,i}\delta_{x_{s\,i}}:\,s\in
S\right\\}\right),$

i. e. $OS_{f}(h)(\mu,\,0)=OS_{f}(h_{0})(\mu)$ and
$OS_{f}(h)(\mu,\,1)=OS_{f}(h_{1})(\mu)$ for each $\mu\in OS_{f}(X)$. In other
words, $OS_{f}(h)$ is the homotopy connecting $OS_{f}(h_{0})$ and
$OS_{f}(h_{1})$ maps. Thus, the functor $OS_{f}$ preserves the homotopy of
maps.

$\scriptstyle\blacksquare$

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