Empirical frame hyper-surfaces as models of multi-parametrical technological systems

This paper is devoted to the mathematical and computer simulation of multi-parameter technological systems. The method and algorithm can easily be used in spline-frame simulation of the systems. Simulation is based on experimental data and achieved by the variation of one-dimensional spline approximations. A set of variable one-dimensional cubic spline-frames generate the model of hyper-surface, which is a model of the process. Each of the spline-frames is the image of a section of input parameters area. Software realization is based on the single algorithm that is used repeatedly. The problem of building some model of hyper-surface is based on empirical irregular 0-dimensional frame. We consider the problem as an inverse problem of modeling. The method has been used in investigation some technological conditions of thread seams for sewing industry. We investigated the durability and harshness of the seam. Geometric models and parameter valuations were generated by special software. The equations and diagrams described our experiments with sufficient exactness and there were used for researching the process. The model together with software HYPER-DESCENT may be applied for simulation multi-parametrical systems or technological processes of light industry.


Introduction
Technological processes in various fields of industry are always considered as multi-parametrical systems. It is impossible to research such systems and processes without correct mathematical models. There are many mathematical and geometric methods of modeling, but it is usually difficult to find a proper model for given multi-parametrical process. Therefore, the quest for new models and development of existing ones are an actual problem at present. Some new promising resources are opened thanks to the development of computational technologies and computer visualization [1]. Traditional approaches to modeling the systems presuppose availability the following data: 1. Some multi-dimensional area of input parameter variations. 2. Allocation the nets of input parameters inside the area. 3. Selection principles for rather simple and adequate functions from some classes. If all these functions circumscribe the output parameters with given precision, they may be considered as models of the system. The nets of input parameters are usually considered as point and regular nets. However, to find a proper model it is necessary to solve a lot of problems. For example, there exists the problem of selection. To solve this problem it is essential to have maximum information about technological essence of the process a priory. If the system or the process belongs to multiparametrical one the problem of preliminary data analyses arises and it is rather hard to solve it formally. If several models describe the same system or the same process it is necessary to choose an optimal model by means a certain criterion [2 -4]. These problems and difficulties force researchers to find other approaches. Approach which is proposed in the paper has the purpose to avoid partly all these difficulties. The suggested method refers to frame modeling of multi-dimensional surfaces since it is based on theoretically developed spline approximation. However after introducing slight modifications it can be used for other geometric objects, too.

Object of research
As an object of investigation we chose thread seams which were produced by shuttle stitches on the sewing machine DDL -8100e Juke. The pressure of working part and diameter of sewing needle (№ 90) were constant. Patterns of stitch seams were executed by sewing threads 25LL, 35LL, 45LL, 70 LL. The seams were carried out along and across the base threats, and at 45 0 angles to the base threats. The numbers of stitches were 2 -3, 3 -4, 5 -6 by 1 sm. Allowance size was equal to 10 mm by the threat. Fabric characteristics were as follows: 1. Filamentous compositions were: NPF on the base and PR/cotton on the others. 2. Surface thickness was equal to 213 g/sm 2 . 3. Weave was linen. 4. Durability (P) on the base was equal to 180 kg. 5. Harshness (EI) on the base was equal to 17764 mcNsm 2 . All properties of sewing threads corresponded to standards. Seam durability was examined by means of breaking machine PT -250M and harshness was examined by apparatus PT -2 according to standards. Geometric models and parameter valuations were generated by special software [5]. The equations and diagrams described our experiments with sufficient exactness and there were used for researching the process.

General considerations
In this paper we consider (n + k)-dimensional Euclidean space X n  Y 1 1  …  Y 1 k together with the following four objects: 1. The hyper-plane X n = X 1 1  …  X 1 n of input parameters. 3. An area E k  Y 1 k , E k = {y k  R: y k, min  y k  y k, max } of output parameter variations. 4. Some hyper-surfaces y k = f k (x 1 , …, x n ). The space of input parameters D is usually considered together with a discrete regular point nets x i, min = x i, 1 < x i, 2 < … < x i, m(i) -1 < x i, m(i) = x i, max , where all numerical values m(i) may be depend upon the values of i. The space of output parameters E is determined by means of experimental data and it is unknown beforehand. Extreme values y k, min and y k, max are determined as a result of decision an extreme problem. The hyper-surfaces y k = f k (x 1 , …, x n ) are given as a discrete sets of points y j(1), …, j(n), k = f k (x 1, j(1) , …, x n, j(n) ), 1  j(i)  m(i). These points form irregular nets in D  E k . Analytical models of hyper-surfaces are usually generated by approximation of input data [6,7]. In our case we may consider the problem of approximation as a problem of regularization and also as researching of some regular mapping D  D  E k . This mapping transforms the irregular net to a regular one.

Direct and inverse problems of modeling
Let's assume that hyper-surface y = f (x 1 , …, x n ) is given analytically in the space D. We may consider the hyper-surface as regular one. Then we can define it by means of some discrete p-dimensional frame, 0  p  n -1. We must have in mind that there is infinite number of modes to build the frame for each value of p. For example, if p = n -1 we may build the following frames: y = f (x 1 , …, x n -1 , a 1 (x n ), a 2 (x n ), …), y = f (x 1 , …, x n -2 , x n , a 1 (x n -1 ), a 2 (x n -1 ), …), . y = f (x 2 , …, x n , a 1 (x 1 ), a 2 (x 1 ), …). These frames are (n -1)-dimensional level surfaces. If p = n -2 we may receive the following frames: y = f (x 1 , …, x n -2 , a 1 (x n -1 , x n ), a 2 (x n -1 , x n ), …), ………………………………….. y = f (x 3 , …, x n , a 1 (x 1 , x 2 ), a 2 (x 1 , x 2 ), …). And so on. We consider the simulation of some hyper-surface by means of p-dimensional frames as a direct problem. It is characterized by the following properties: 1. The property of regularity for p-dimensional frame, 0  p  n -1. 2. The frame may be tightened up to infinity. 3. There exists the possibility to diminish the space of input parameters. If we can to describe p-dimensional frame as an (np)-parametric family of p-surfaces of the same degree, we have a regular frame. Let's assume that hyper-surface y = f (x 1 , …, x n ) is given by irregular net of experimental points in the space D  E. This net may be considered as empirical 0-dimensional frame. Simulation of multi parametrical technological processes compels us to deal with precisely such frames. The empirical frames having the properties of regularity are not typical ones into practice. Fig. 1 shows the differences between some theoretical regular frame and empirical irregular one. All attempts to stretch some regular surface on an empirical frame or, in other words, to find some regular mapping lead to considerable complexity of the model or to the great loss of precision. Generally speaking, it leads to loss the sense of modeling, at all. Inverse problem of creating some model of hyper-surface arises when we want to use an empirical irregular 0-dimensional frame. It is necessary to take into account the fact that irregularity of data may be revealed in any dimension from p = 1 up to p = n -1. Here, we consider only one case when irregularity is appeared if p = 1. It means that one can build the model of hyper-surface in form of some 1-dimensional frame, but it is impossible to build any regular 2-dimensional frame. Irregular 1 dimensional frame is usually built as a cubic spline net. Preliminary evaluation of regularity is fulfilled visually.

Conclusions
The mathematical simulator which is an empirical one-dimensional spline-frame of hyper-surface inside the area of parameters is described. Effectiveness of the model is confirmed by applications to sewing industry. The model allows us to solve some similar problems for the other fields of industry. The model together with software HYPER-DESCENT may be applied for simulation multiparametrical systems or technological processes of light industry. Software HYPER-DESCENT uses only one-dimensional spline approximation. This approach simplifies the process of the software creation. Evaluation of regularity of the frames is realized visually by means of outputting appropriate diagrams.