=Paper=
{{Paper
|id=Vol-3027/paper75
|storemode=property
|title=Predictive Geometric Models for Heat-Insulation Properties of Semi-Finished Fur and Down Products
|pdfUrl=https://ceur-ws.org/Vol-3027/paper75.pdf
|volume=Vol-3027
|authors=Victor Yurkov,Elena Dolgova,Margarita Chizhik
}}
==Predictive Geometric Models for Heat-Insulation Properties of Semi-Finished Fur and Down Products==
https://ceur-ws.org/Vol-3027/paper75.pdf
Predictive Geometric Models for Heat-Insulation Properties of
Semi-Finished Fur and Down Products
Victor Yurkov 1,2, Elena Dolgova 1 and Margarita Chizhik 1
1
Omsk State Technical University, 11 Mira Prospekt, Omsk, 644011, Russia
2
Omsk State Pedagogical University, 14 Tuhachevsky Embankment, Omsk, 644099, Russia
Abstract
This paper is devoted to geometric simulation of heat-insulation properties of fur and down
products which are considered as multi-parameter and multi-component systems. We consider
predictive models of heat resistance depended on physical characteristics of fur and pelt. There
is a problem of construction co-ordinate geometric models on condition that the set of
experimental data is limited. We solve the problem as a problem for static multi-component
systems. The model is considered as a piecewise constant function in the space of input and
output parameters. The paper proposes an algorithm of construction the clusters on the set of
given experimental points. Moreover, we construct multidimensional convex covering on the
set of the points. The covering is based on its two-dimensional projections. Results of the
investigations allow us to substantiate producer’s choice of fur and down semi-finished
products and its composition for manufacturing the product of special purpose. The method
suggested in the paper may be one of geometric modulus of the software HYPER-DESCENT
which has been developed formerly. Our geometric models together with software HYPER-
DESCENT may be applied for simulation and prediction the properties of another multi-
parametrical systems or technological processes of light industry.
Keywords 1
Multicomponent system, fur semi-finished product, heat-protective properties, geometric
modeling, clustering.
1. Introduction
At present there are many problems connecting with technical control and prediction various
properties of fur and down products. Real fur and fur clothes have high heat-insulation properties. All
properties are conditioned by slight air permeability of leather and high heat resistance of hair covering
that causes an air layer. All heat-insulation properties of fur semi-finished products depend on the length
and density of the hair and also these properties depend on thickness and density of the leather. That
leads not only to various versions of constructive models but to various combinations of materials in
the same heat-insulation multi-layer package. At present all classifications of heat-insulation properties
for fur and down semi-finished products are relative ones because heat resistance depends not only on
properties of the pelt but it depends on the breed, quality of the pelt, finishing operations and so on.
Therefore, elaboration of new methods and design of predictive geometric models for fur and down
products are actual problems at present.
As for geometric aspect of the problem it is necessary to take into account the following
circumstances. Firstly, there is a problem of multi-factorial nature of objects. A great number of factors
are inaccessible and indeterminate. Hence, we have a problem of data analyses. Secondly, one can be
faced with different experimental output results even if all input parameters are the same ones. It means
existence of some unknown factors. Thirdly, some of input parameters may be fuzzy. And at last if we
want to find a proper predictive model as a monoidal hyper-surface in the space of input and output
GraphiCon 2021: 31st International Conference on Computer Graphics and Vision, September 27-30, 2021, Nizhny Novgorod, Russia
EMAIL: viktor_yurkov@mail.ru (V. Yurkov); dolgova13@rambler.ru (E. Dolgova); margarita-chizhik@rambler.ru (M. Chizhik)
ORCID: 0000-0003-2667-8103 (V. Yurkov); 0000-0003-3174-7142 (E. Dolgova); 0000-0003-0797-875X (M. Chizhik)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
parameters we must have a great number of experimental samples given by an appointed plan. But it is
impossible because it is impossible to rear the animals with beforehand given properties of their fur and
pelt.
Limited number of experimental data requires realization of original modeling approachs. We
analyze all data and appraise possibility of separation the data into clusters. Then we simulate properties
of the clusters by means of piecewise constant functions. As we accumulate enough number of data we
may simulate some cluster properties by polynomials. Finally, we may find the same piecewise model
for input parameters area as a single whole [1, 2].
2. Object and aim of the researching
The aim of this paper is to give a description on analytic representation of heat-insulation properties
for fur and down products.
As an object of investigation we were given several samples of rabbit pelt only. Physical parameters
of the pelt are in Tabl. 1.
Table 1
Mass and geometric characteristics of rabbit pelt
Thickness of leather, Length of the fue Density of the leather, Density of the hair cover,
mm hair, mm g/cm3 g/cm3
0.95 ± 0.1 6 ± 0.5 1.50 ± 0.2 0.056 ± 0.005
0.43 ± 0.05 9 ± 1.0 0.69 ± 0.07 0.048 ± 0.005
0.53 ± 0.05 25 ± 2.5 0.63 ± 0.07 0.025 ± 0.005
0.59 ± 0.05 15 ± 1.5 0.66 ± 0.07 0.033 ± 0.005
0.51 ± 0.05 10 ± 1.0 0.48 ± 0.05 0.040 ± 0.005
0.75 ± 0.07 9 ± 1.0 0.47 ± 0.05 0.033 ± 0.005
0.51 ± 0.05 5 ± 0.5 0.61 ± 0.07 0.027 ± 0.005
3. General theoretical considerations
In this paper we consider one static mathematical model of multi-component system [3, 4]. We are
limited both in the number of input parameter realizations and in corresponding realizations of output
parameters. Limited sets of input and output parameters are considered as initial data of modeling. Let
X be a set of input parameters and let Y be a set of output parameters of the system. Our purpose is to
identify a nonlinear correspondence Y = F(X, A). Specifically, we have: dim X = n, dim Y = 1.
Special feature of the problem is that all realizations of parameters are random. The structure of the
model may be chosen subjectively or it may be identified objectively. If the structure has m numerical
parameters we need m random realizations at the minimum. Moreover, preliminary structure analyses
of geometric data may be at variance with a formal mathematical model. For example, structural
contradiction of data and formal model is shown in Figure 1. To eliminate the contradiction one can use
a piecewise constant model which is shown by dash lines. Using a more complicated model we take a
risk to have an absurd result.
If the number of parameters for linear model exceeds the number of given realizations we have some
indetermination of the model. And at last geometric structure of data may be unfit for extrapolation.
That takes place if values of input parameters have comparatively small differences but corresponding
values of output parameters have a great differences.
We begin by consideration the space X Y, dim (X Y) = n + 1, having a rectangular n-dimensional
area X: ai xi bi. i = 1, …, n, and a section Y: ymin y ymax. Probable locations of the area X and the
section Y are determined by a priory information and physical meaning of data. In X we have finite set
P of points Pi, i = 1, …, p. Each point Pi P is considered as a centre of some n-dimensional rectangular
subset (x1 a1, x2 a2, …, xn an)i. The number of points Pi may be lacking for finding a polynomial
model and all locations of the points may be chaotic in X. All given points may generate some compact
groups or they may be located more or less evenly. Being limited by all these conditions we want to
find an analytic model y = f(x1, …, xn) of the system and also we need possibility of extrapolation the
values of y onto the points which will be given out of the set P.
In order to solve the problem we suggest several stages. At first stage we construct a
multidimensional convex covering of the finite set P [5]. First principals of covering are illustrated by
Figure 2.
y y
x x
Figure 1: The model is in accordance with the data (on left side) and the model is not in accordance
with the data (on right side)
x
x1j + b1j
P1
X P3
ij
P2
x2j – b2j
x
x1i – a1i x3i + a3i
i
Figure 2: Two-dimensional convex covering Xij of the set P1, P2, P3
We have:
P1(x1i a1i, x1j b1j), P2(x2i a2i, x2j b2j), P3(x3i a3i, x3j b3j),
Xij = {x1i – a1i xi x3i + a3i} {x2j – b2j xj x1j + b1j}
{xj f(x1i + a1i, x1j + b1j, x3i + a3i, x3j + b3j,)}
{xi f(x1i – a1i, x1j – b1j, x2i – a2i, x2j – b2j,)} {xi f(x2i + a2i, x2j – b2j, x3i + a3i, x3j – b3j)}.
Here Xij is a projection of the covering onto the plane Oxixj. All combinations of two-dimensional
projections of the set P and all their convex coverings are constructed by the conditions i = 1, …, n –
1; j = 2, …, n; j > i. As a result we have n-dimensional convex covering X of the set P: X = Xij.
The second stage includes some data clustering algorithms [6, 7]. All algorithms are based on
supposition that geometric nearness of points Pi and Pj, i j, means a physical closeness of its originals.
Complete number of clusters is unknown beforehand. It may be in interval from 1 up to p. Nearness of
points is considered as a distance d(Pi, Pj) in Euclidean or Hamming metrics.
As for the space Y it may be divided into strata Yk = {yk: ykmin yk ykmax}, 1 i p. In general case
we have Yk Y and a number of strata may be in interval from 1 up to p. Points Pi P are united at
one cluster if ykmin yk (Pi) ykmax. After that we verify connexity of the clusters. Any of clusters is
considered as a connected one if all its points Pi are in the same stratum. If it is not so the cluster is
considered as a disconnected one and it is divided in two or more parts that are new clusters.
At the third stage we divide the area X into sub-areas having closeness of their physical parameters.
Since the number and location of all clusters are known it is necessary to find geometric place of points
which distance to given cluster is less then distances to all another clusters. In other words we have to
construct some similarity of Voronoi diagram of order k in multidimensional space. That problem is
connected with considerable algorithmic difficulties and that is why we prefer more simple
approximating algorithms. Its principle is shown in Figure 3.
x
j d
12 2
d
1 12
d
23
d
d d 23
13 13
3
x
i
Figure 3: Construction of separating hyper-planes for the clusters 1, 2 and 3 in Hamming metric
We construct convex envelopes for all clusters. Each convex envelope is a closed broken line, if n =
2, or it is a multidimensional closed polyhedron, if n > 2. Separation of one envelope from another is
realized by hyper-planes. The set of separating hyper-planes consists of hyper-planes which are parallel
to coordinate (n – i)-dimensional spaces, where i = 1, …, n – 1. For example, all separating hyper-
planes shown in Fig. 3 are hyper-planes perpendicular to coordinate plane Oxixj. The hyper-planes
which separate two neighbouring clusters are construct as hyper-planes equidistant to these clusters.
Our fourth stage includes the construction of piecewise constant data approximation. Constant
function
yk = yi / pk, Pi(x1i, …, xni, yi) Xk Yk, i = 1, …, pk
is constructed in each stratum. To illustrate constructive principle some conventional monotonous
piecewise constant model which approximate the data for three connected clusters and three separated
strata Y1, Y2, Y3 is shown in Figure 4.
y
Y3
Y2 x2
Y1 F2(x1,x2)
F1(x1,x2)
x1
Figure 4: Piecewise constant approximation of three clusters and three strata
4. Method of research, fabrics and equipments
Total heat-insulation was determined by original method which has been developed and patented
beforehand. The gist of the method was the following (see Figure 5). A heat accumulator was placed
into the packet made of fur and down semi-finished materials. The temperature was varied purposefully.
The moment of heat accumulator cooling was recorded. Calculation of total heat-insulation was
computed by formula R = (S t) / (C m), where S is an area of the sample, t is total time of cooling, c
is a heat capacity of the sample and m is a mass of accumulator. The results are shown in Tabl. 2.
Table 2
The values of total heat resistance
Time of cooling, s Heat resistance, m2K/Wt
5160 0.681
4680 0.617
4800 0.634
4740 0.625
4560 0.602
4320 0.570
3660 0.483
Thickness of the leather was measured by thickness gauge and the length of the hair was measured
by a special measuring bar.
Original device for heat-insulation measuring consists of heat accumulator which has a special gel
and hermetic container. The latter is a hollow cylinder and the sample is wound round it. The sample
consists of one or several pelt strips. The length of the pelt strips is varied from 186 up to 378 mm. The
strip is from 20 up to 140 mm wide.
Electric
Thermocouples force gauge
Heat
accumulator
Ceramic
Mounting
heating
Heat- appliance
insulation
package
Sample
Figure 5: The device for total heat resistance determination
5. Predictive model for heat resistance
Let x be a thickness of the leather, y be a length of the hair, z be a density of leather texture, v be a
closeness of hair covering and t be a heat resistance. After experimental researches of all given fur and
down samples we have the following mathematical model of heat resistance. Four-dimensional
covering X = Xxy Xxz Xxv Xyz Xyv Xzv of the set P is described by the following inequalities
Xxy : 5 y 25; 150x – 50 y –50x + 55; 0.4 x 1;
Xxz : 0.4 x 1; 0.4 z 1.5; z 1.6x + 0.06;
Xxv : 0.4 x 1; 0.02 v 0.04; (4/30)x – 2.2/30 v (1/60)x + 2.6/60;
Xyz : 5 y 25; 0.4 z 1.5; z – (0.8/19)y + 28.98/19;
Xyv : 5 y 25; 0.02 v – (0.03/20)y + 1.35/20;
Xzv : 0.4 z 1.5; 0.02 v; (1/30)z v (1/110)z + 5.1/110.
Formal clustering algorithm gives us the following results CL1 = {P1}, CL2 = {P2, P4, P5, P6}, CL3
= {P3}, CL4 = {P7}.
The corresponding coverings are the following
X1 : x > 0.7; y < 7; z > 1; v > 0.04; x/1.3 + v/0.09 > 1;
X2 : 7 < y < 20; x/1.3 + v/0.09 < 1; x/0.75 + v/0.057 > 1; z < 1; v > 0.03;
X3 : y > 20; z < 1; x/0.75 + v/0.057 < 1; v < 0.03;
X4 : x < 0.7; y < 7; z < 1; x/0.75 + v/0.057 < 1; v < 0.03.
Heat resistances are
t(X1) = 0.681 0.07; t(X2) = 0.6 0.06; t(X3) = 0.634 0.06; t(X4) = 0.483 0.05.
6. Conclusions
We have obtained the piecewise constant analytical model of heat-insulation for fur and down semi-
finished products. Heat resistance of the products depends upon superficial physical characteristics of
fur and pelt. The physical characteristics are partly determined by thickness and density of the leather,
length and closeness of the hair covering.
Using the investigations we can motivate our selection of components and the structure of fur and
down fabrics which are used for manufacturing the special products. Using the analytical descriptions
of heat-insulation we can forecast its values before manufacturing [8, 9]. The method of predictive
geometric models may be one of the modulus of software HYPER-DESCENT and it may allow us to
solve the predictive problems [10].
Throughout the paper we have required the piecewise regularity of the model. Taking that into
account we may note the partial solution of the problem of searching optimal correlation between
structure properties of fur and down pelt and heat-insulation properties of the products. We think that
general solution of the problem needs further investigation. Another direction in which the theory could
be generalized is to investigate the problem of designing multi-component products having beforehand
given properties.
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