Data-measuring device to determine the thermophysical properties of polymer composites: information support part

This paper deals with the problem of developing an information support subsystem of the data-measuring device for determining the thermophysical properties of polymer composites during cure, designed to identify the parameters of a mathematical model necessary to optimize the curing process of composites. Methods and algorithms for processing experimental data and calculating the thermophysical properties of polymer composites, based on an integral representation of the solution of the inverse heat conduction problem and a Kalman filter, has been developed. The structure of the construction, content, goals, parameters and functionality of the basic modules of the data-measuring device application software are presented. A subsystem of informational decision-making support has been developed for choosing the optimal method and algorithm for calculating the thermophysical properties of polymer composites, which allows minimization of the errors in determining properties under given experimental conditions. The subsystem is built on the alternative use of four algorithms for calculating thermophysical properties and is included as a module into the application software of an intelligent data-measuring device for studying the process of curing polymer composites.


Introduction
In recent years, traditional materials such as metals, wood, concrete and others have practically exhausted their capabilities in various industries. They are replaced by artificial materials created with a specific set of properties required for the needs of a specific field of technology. These materials at the present stage of development of technology are polymer composites (PC), which are promising structural materials [1]. They are a combination of a polymer matrix (resin) and reinforcing filler (glass, carbon or synthetic fiber, fabric or roving) and possess a combination of unique physicochemical properties.
The field of application of polymer composites is expanding more and more, giving the industry the task of intensifying the production of composites, as well as developing new and improving the characteristics of existing PCs. Particularly relevant is the problem of obtaining new improved materials for such industries as aviation and space. In this regard, more than ever important is the study of not only the properties of finished materials, but the characteristics of semi-finished products in the manufacturing process, as well as production conditions.
The solution to the problem of studying the properties of materials is impossible without the use of measuring equipment. Control and measuring operations have long been an integral part of technological processes of production and significantly affect the quality of products.
The properties and quality of polymer composite products depend largely on the ingredients and are determined by the temperature-time curing cycle [1]. Therefore, it is necessary to calculate the optimal technological curing cycles of PC products.
Calculation and optimization of PC curing cycles involves the use of mathematical models and determination of the parameters of these models [2]. The important parameters of the PC curing model are thermophysical properties (TPP), both in the curing process and cured PCs. At the same time, the accuracy of determining TPP PC has a significant impact on the further use of the studied characteristics when calculating the optimal curing cycle for PC products [2][3][4][5][6][7][8][9][10].
The polymer composites thermophysical properties that is volume heat capacity ) , , (   T C and thermal conductivity ) , , (    T are nonlinear functions of temperature T, the degree of cure  , and resin content  . The TPP of composites in the cured state, as well as other materials, can be determined in any conditions, by any method suitable for use. On the contrary, when determining TPP PC during the curing process, it is required to reproduce the temperature-time conditions corresponding to the curing cycle. Thus, taking into account the specificity of the PC, it is necessary to conduct their research in conditions close to production, starting the study from the initial state of the PC -the prepreg to its final cured state -the composite [7][8][9][10][11][12]. In this case, these properties become in a certain sense effective, taking into account all the features of the curing process. Therefore, the actual task and purpose of this work is the development of mathematical, algorithmic and experimental support for determining the TPP PC in the process of heating and curing.

Supporting subsystems 2.1 Mathematical support
In modern measuring technology, a large number of different methods and devices have been developed for determining the TPP of materials. They have their own characteristics and are intended mainly for a certain class of materials. However, there is no universal method and device for determining the TPP that operates in a wide temperature range and under any experimental conditions, including the curing process and the determination of TPP both in cured PCs and during cure.
Methods for determining the thermophysical properties of materials are based, as a rule, on solving the inverse heat conduction problem [13][14][15]. The initial data for them are the temperature field and heat fluxes. The results of solving the inverse problems are heat capacity, thermal conductivity, thermal diffusivity and other properties. They give good results for constant and linearly varying TPPs, and have complex solutions for nonlinear TPP [15]. Therefore, in the instrumental implementation of these methods, we try to create such boundary conditions for experimental studies that make it possible to obtain simple calculation formulas and algorithms for processing experimental data. In this regard, when studying the TPP PC, especially in the curing process, it is rational to use special samples and experimental setups that make it possible to simulate the technological curing cycles, as well as to ensure the creation and maintenance of special boundary conditions. At performing TPP studies, it is advisable to use the temperature distributed over the thickness of a flat sample or on its surfaces as initial data. This can be realized by heating the sample on one side and thermal insulation of the other surface of the sample. Such heating allows halving the consumption of the studied materials. A special requirement when conducting a ТPP PC study is to exclude the flow of the resin in the direction of the temperature gradient. If possible, the complete elimination of the resin flow is also desirable. In the case when it is impossible to completely eliminate the course of the resin, it is possible to admit the presence of the flow of the resin along the isotherm. Then, in this case, the method of determining the TPP should take into account the change in the resin content  and sample thickness ) (t L . When solving the inverse heat conduction problem, it is convenient to use the values of heat fluxes 0 q , L q on the surfaces of the sample under study as input data for the calculation. In this regard, when constructing a mathematical model of heating and curing PC, the boundary conditions of the second kind will be set.
For the experimental determination of TPP PC, a mathematical model of organizing the process of heating and curing the studied samples of polymer composites of the following type was proposed: where C is the volume heat capacity, J/(m 3 ·K); 0 f is the initial temperature distribution, K; 0 q , L q is the heat flux density on the sample surfaces, W/m 2 ; L is the sample thickness, m; T is temperature, K; t is the time, s; m t is the duration of the experiment, s; x is the spatial coordinate, m;  is thermal conductivity, W/(m·K).
In the mathematical model of heating and curing of the polymer composite sample, the unknown and the quantities of interest to us are the thermophysical properties, which are the heat capacity ) (T C and thermal conductivity ) (T  , depending on temperature T . Additional to the presented non-linear model, it is also possible that the TPP are constants C ,  or functions of time x . The number of points j in time is limited only by the amount of information received. At the same time, the number of points i in the spatial coordinate is limited by the possibility and the number of establishing thermocouples across the thickness inside the sample, the embedding of which will not distort the temperature field ) , (  t  x  T . Based on the transformation of the mathematical model (1), a set of methods for processing experimental data and methods for calculating the TPP of a PC has been developed.
Consider the general principles and results of the construction of methods for determining TPP.
In the first method of calculating the TPP, we used the integral transformation of the model equation and the assumption about the time dependence of the TPP with correlation of time t to the temperature ) (t T . The main calculation of TPP is performed according to the following formulas: The stability of the numerical differentiation operation is achieved by smoothing the experimental temperature distribution ) , ( using cubic splines. To calculate the values of the integrals in formulas (2), (3), we used an apparatus of quadrature formulas, which gives the best results of the calculation in the presence of three or more experimental points along the spatial coordinate x over the thickness of the sample.
In the second method of calculating the TPP, we also used the integral transformation of the model equation and the assumption about the time dependence of the TPP, but the calculation formulas are adapted for the case when temperature measurements are made on only two surfaces of the sample ) ( 0 t T and ) (t T L . This is due to the fact that in the practice of thermophysical measurements it is often not possible to measure the temperature inside the sample through its thickness. In such cases, we are limited to measuring the temperature only on the surfaces of the sample ) ( 0 t T and ) (t T L . Since in the quasistationary mode, the temperature distribution across the thickness of the sample has the shape of a parabola and is accordingly determined by the dependence , it is not enough for two points of temperature on the surface to restore its function. When constructing a parabolic approximation, it is also necessary to take into account the values of heat fluxes. As a result, we obtain the following calculation formulas: The calculated formulas (2), (3), (4), (5) allow us to determine the time dependences of the heat capacity ) (t C and thermal conductivity ) (t  , referred to the average integral temperature or their constants C and  . When there is a significant temperature dependence of TPP, then the resulting calculation formulas have an increased error and can not be used to determine the TPP PC, especially in the curing process. Therefore, there is a need to develop more complex methods and algorithms to determine the temperature dependences of TPP. The third method is based on an integral representation of the solution of the inverse heat conduction problem in the form of an integro-functional equation and the construction of an iterative procedure [15]. The result of the solution are the functions ) As the initial approximation of heat capacity C and thermal conductivity  , we used the values calculated by formulas (2), (3) or (4), (5). Further calculations are made according to the following iterative procedures [15]: is calculated by formula (6) using the found values of the initial approximation of the volume heat capacity according to (2) and the experimentally measured temperature distribution and then it is used in the calculation of the iterative procedure of (7). Performing the iteration process (7), after each iteration of calculating At The calculated values of the volume heat capacity are then used to determine the thermal conductivity.
At the beginning of calculations, using formula (3) or (5), we find the thermal conductivity value )) , referred to the average integral temperature, then we determine the thermal conductivity corresponding to the temperature on the sample surfaces )) ( , by the equation: The fourth method is based on the use of the Kalman filter, which allows by the mathematical model of the heat transfer process to construct the optimal TPP estimates via the experimental measurements of the temperature distribution ) , ( . Using the linear Kalman filter and the formula for the analytical solution of the heat conduction equation with boundary conditions of the second kind for the heating problem on one side of a plate with thickness L in a quasi-stationary mode, we obtain an iterative procedure for calculating the heat capacity and heat conductivity: where j T is the experimental temperature field for the plate; j T is the estimate of the temperature field in discrete time; 0 D is the dispersion of noise, which is determined by the accuracy of the measuring device.
As a result, a recursive filter was obtained, which allows evaluating TPP of the sample from a series of inaccurate noisy measurements of the temperature distribution in it, moreover, the method allows determining TPP from significantly perturbed experimental data and is stable when processing any experiments.
Thus, a complex of methods for processing experimental data and calculating the TPP of polymer composites based on an integral representation of the solution of the inverse problem of heat conduction and using the Kalman filter, has been developed.
Based on these methods, four algorithms were developed for processing experimental data and calculating thermophysical properties. However, each method and algorithm developed by us has some specific features and limitations on its use. In addition, the accuracy of the results obtained and the stability of the calculations for each algorithm differs when the conditions of the experiments are changed. For this reason, it is impossible to find a universal algorithm that would provides for minimum errors in the calculation of the TPP regardless of the experiment conditions. Therefore, when determining TPP, we need to conduct numerical processing of the experimental results using the available algorithms, compare the results of calculating TPP and choose one of them, based on a priori data. The delivered task is nontrivial, since the comparison of the results of the algorithms requires the presence of an expert who is able to assess the compliance of the calculated TPP with real values. Additionally, this process can be lengthy and does not guarantee the successful solution of the problem of determining TPP. These arguments make it necessary to automate the decision-making process for choosing an algorithm which provides minimum errors in the calculation of TPP PC.
Therefore, it has became necessary to develop mathematical, experimental and algorithmic support, built on the basis of the developed methods for calculating TPPs that, by analyzing the initial conditions of the experiment, can give the user recommendations on choosing the optimal method, which has minimal errors in determining the TPP during heating and curing of the polymer composites.

Experimental support
To study the parameters of the curing process of polymer composites as a whole and to determine their TPP, we developed a data-measuring device (DMD), a detailed description of which and the stages of its development are presented in [16][17][18][19].
The DMD allows performing the following functions: communication of the measuring instrument with the object of study; registration and conversion of information; numerical processing of experimental data (primary data processing, that is, preliminary processing of the initial experimental data, and secondary processing, that is, the calculation of the desired parameters of the material under study based on the solution of inverse problems); visual presentation and preservation of the initial conditions of the experiment and the results of processing; regulation and maintenance of the specified research conditions; experiment planning; management of the experiment; results documentation; computer simulation of the experiment; checking the adequacy of the experimentally studied parameters of the proposed mathematical model.
The generalized structure of supporting DMD subsystems for studying the properties of PCs, their components, and the relationships between them is shown in Fig. 1 and includes:  mathematical support, which is a set of mathematical models, methods and algorithms;  software, which is a set of all software procedures, system and application software;  information support, which are methods of information display and storage of data on the state of the object under study for further use in the system and the central part of the information support is the database;  technical support, which is a set of technical means and a measuring instrument;  organizational and methodological support, which is a set of documents establishing the composition of the DMD, operating rules, etc. The software of DMD for determining TPP PC includes system software and application software. The application software that forms the basis of DMD for determining TPP PCs is a set of software modules.The integrated visual development environment Delphi 7 was chosen as a tool for creating the application software, which ensures the rapid development of various software applications, the convenience of which lies in the presence of many built in ready to use components for various programming areas and user friendly interface.
The application software includes five modules:  control module, it provides input in the dialogue mode of information about experiment, the management of the experiment, the acquisition and saving records of experimental and calculated information in the database;  calculation module of ТРР РС, it performs calculations based on the initial data received from the control module of experiment [18,19];  simulation module, it performs model simulation experiments on testing algorithms for calculating the thermophysical properties of PC;  decision support module (DSM) for choosing the optimal calculation method, based on the data on the results of the experiment, it allows you to choose the best method and algorithm for calculating the TPP, which provides the minimum methodological error;  information visualization module, it provides a graphical representation of experimental and calculated data.
In accordance with this purpose, the mathematical formalism was developed for DMD comprising four methods determine TPP on the basis of which we proposed:  algorithm #1 for calculating TPP as a function of time by temperature inside the sample;  algorithm #2 for calculating TPP as a function of time by temperature on the sample surfaces;  algorithm #3 for calculating TPP as a function of temperature based on the integral transformation of the inverse heat conduction problem in the form of solving an integrofunctional equation for temperatures inside the sample [15];  algorithm #4 for calculating TPP as a function of time based on Kalman filter.
The application software has been developed on the basis of algorithmic support of the DMD, which is part of the module for calculating thermophysical properties.
To choose the best algorithm for calculating TPPs, a decision-making support module was developed that significantly helps the DMD user correctly process experimental data. After choosing the algorithm, TPPs are calculated and visually displayed in the form of graphs and tables [16][17][18][19][20][21]. The calculation of TPP by all algorithms at once is also provided. Visualization of calculations can also be displayed on a single graph for the possibility of comparing the results.

Simulation support
The primary task in determining TPP is to obtain the desired values with minimal errors. For this reason, the problem arises of choosing the best algorithm for processing experimental data and calculating TPP. Therefore, the main criteria for evaluating the calculation algorithms are the errors and the possibility of their application. Analytical and numerical methods can be used as methods for estimating errors and testing the stability of solving inverse problems in processing noisy input data [15,22]. Analytical methods for estimating errors are distinguished by the complexity of the calculation formulas and equations that makes it very difficult, and in some cases makes it impossible to perform a complete analysis, which is possible in the case of using numerical methods. The use of numerical methods makes it possible to obtain the most accurate correspondence between the real and accepted mathematical formulations of problems. Therefore, the use of numerical methods is most convenient for studying the methodological errors of complex calculation formulas, algorithms, and equations for solving the inverse heat conduction problems in comparison with the analytical solution of direct problems [15].
Thus, to estimate the errors in determining TPP using one or another algorithm in specific experimental conditions, it is necessary to build a subsystem that allows simulating various experimental conditions and determining the errors of the TPP calculated under these conditions.
Since the error in the calculation using any of the algorithms varies depending on the experimental conditions, and the fact that the algorithm can be applied also changes, then it is impossible to unambiguously recommend only one algorithm of calculating the TPP for processing all possible experiments. The choice of algorithm for calculating TPP will depend on the conditions of the experiment. Therefore, it is necessary to evaluate the conditions of applicability for each of the developed algorithms, including performing:  analytical study of the algorithm for calculating the TPP to identify constraints for its application, explicitly resulting from the calculation algorithm;  simulation modeling to determine the dependence of the errors in the calculation of TPP on the input conditions of the experiment. The methodology for the simulation modeling includes the following stages. At the beginning, the type of temperature dependence of TPP is chosen, which should be similar to the properties of polymer composites, including the dependence of TPP in the curing process. After, the approximating function for TPP is introduced by setting the corresponding coefficients of the polynomial and the remaining experimental parameters. Then, the direct heat conduction problem is solved and the temperature field is found in a discrete form ) , ( . The obtained data is transferred to the TPP calculation subsystem for solving the inverse heat conduction problem. The found TPPs are transferred back to the simulation modeling subsystem and compared with the given TPPs, and then the calculation error of the chosen algorithm is determined [15].
The proposed method of the simulation study of accuracy and assessment of the possibility of using TPP calculation algorithms form the basis of the simulation modeling module when choosing the best TPP calculation algorithm.

Decision-making support
The subsystem of information support for decision-making on the choice of the optimal TPP calculation algorithm is based on a modified hierarchy analysis method designed to solve multicriteria problems with a finite set of possible solutions [23].
The application of the method is based on the representation of the analyzed problem in the form of a hierarchical structure. At each level of the hierarchy and on the basis of expert assessments, the comparison of elements is performed by importance and the results are presented in the form of a matrix of pairwise comparisons. The method is used in cases where it is impossible to quantify all the links between the components of the system and establish these links by expert means. The hierarchical structure of the algorithm choice problem is presented in Fig. 2.
The fundamental complexity of the problems of choice in many criteria lies in the impossibility of a priori determination of the best solution. To build a decision-making model for choosing the TPP calculation method, we will present it in the form of a hierarchical structure: the target is placed on the first level, the criteria composition is placed on the second level, and many alternatives are placed on the third level. Therefore, the essence of the method is to use a matrix of pairwise comparisons, built on the basis of expert data on the relative importance of the criteria [24].
The main aim of the subsystem DSM is calculation the TPP PCs with the minimal methodological error under the specified conditions of the experiment. The aim can be achieved by choosing one of the TPP calculation methods that is one of the alternatives. The choice of the most successful of the alternatives is made on the basis of their comparison with each other by criteria. In our case, as criteria, it is rational to use the list of experimental conditions and evaluate the actual operability of the algorithm and the error depending on the criterion.
Using the DMD simulation module, based on numerous numerical simulation experiments in accordance with the criteria, matrixes of method estimates were constructed. Having obtained the experimentally specific values of the criteria from the matrices, the actual performance and errors of the method are evaluated for each criterion. In order to get a general assessment of the alternative for all the criteria, it is necessary to evaluate the criteria in order of importance and consequently construct a matrix of pairwise comparisons. Based on the analysis and mathematical formalization of the problem, we obtain an analytical model of the decision support process for choosing the optimal calculation method and algorithm:  (12) where Z is the purpose of building DSM;  (13) The vector of the total assessment of alternatives is calculated as: For obtain a matrix for evaluating methods according to several criteria, we perform simulation experiments, pre-determined the list of criteria  The simulation modelling is conduct on the basis of a sequential enumeration of various predicted situations that arise when measuring TPP, by changing the values of one criterion with the same values of other criteria and obtaining calculation errors for all methods. After completion of calculations with a change in the value of one criterion, it is necessary to proceed to the next and so on. As a result, we obtain a set of value ranges for each criterion and a corresponding set of TPP error values calculated by each of the methods. The obtained data are recorded in the form of a set of rules of the knowledge base in the format: Thus, a matrix has been constructed to evaluate the methods of calculating TPPs in accordance with the formulated criteria Kr , using the rules of the knowledge base (15) for specific experimental conditions.
The weight vector of criteria W is determined by constructing a matrix of pairwise comparisons, which is obtained on the basis of expert estimates. One of the criteria is selected, with which it is most convenient to compare all the others. Then the expert determines how many times the weight of the first criterion is greater than the weight of the second. Next, the first criterion is compared with the third, and so on, sequentially. After such comparisons, the numbers

Results and discussion
The stability and errors of the obtained solutions using the developed methods and algorithms are studied on the basis of simulation modeling of experiments on heating and curing PC. The results of the algorithms showed that the errors and stability of the calculated TPP values obtained with their help are different and vary depending on the errors of the initial data. To establish the relationship between the input data of the experiment and the error of the algorithm, a numerical simulation of experiments with successive smooth alternating changes in the input conditions was performed. As a result of analyzing the changes in the errors and stability of the TPP calculation using the described algorithms, performed on the basis of simulation modeling and tested on extensive experimental material, as well as by determining the mathematical and logical limitations of applying calculation methods, data was obtained to build a decision support module of DMD for choosing the best TPP calculation method.
Availability of the software in the simulation module allows us to conduct numerical experiments to compare the developed methods and algorithms, to assess the stability and the possibility of applying these algorithms to calculate TPP for different values of each of the input parameters of the experiment. The simulation study begins with the selection of a list of experimental conditions, the change in the values of which is largely reflected in the TPP calculation. The choice of such parameters is made empirically based on knowledge about the passage of a real experiment. The main input conditions of the experiment, selected for further analysis, were taken: the number of thermocouples for measuring the temperature over the sample thickness, the temperature differential over the sample thickness and the type of TPP functions. The type of TPP dependence on temperature was set as constants, as functions linearly increasing and decreasing in two times, as well as a function with an extremum, imitating the process of curing the composite. The initial data for simulation was the calculated temperature field, on which white noise was applied with amplitude of 0.1 K.
As a result of the analysis of the conducted simulation experiments, a summary table 1 was obtained, which characterizes the relationship between the input conditions of the PC sample heating and curing experiments and the ability to use one of the four TPP calculation algorithms.
The performed simulation studies were used in the development of a decision support module. The DSM automatically makes a decision on the choice of one or another TPP calculation algorithm from the four developed, which has a minimum error and sufficient stability when processing the received data in some experimental conditions, and recommends it to the DMD user when calculating TPP.
The data presented in the table confirm the statement about the absence of a TPP calculation algorithm that is universal for various combinations of experimental input conditions and indicates the presence of limitations in the application of each of the proposed algorithms.
However, for solving a practical problem of increasing the accuracy of calculating the TPP, the choice of the calculation algorithm plays a key role. In addition, the table takes into account the effects of only three basic conditions of the experiment on the possibility of using algorithms. The inclusion of additional parameters in the analysis leads to its substantial complication and the impossibility of forming the final conclusions in the form of a table. Thus, the developed algorithms complement each other's capabilities and expand the area of experimental conditions.

Conclusion
A subsystem of information support for a data-measuring device for studying the PC curing process and determining their thermophysical properties is proposed. The mathematical support for the definition of TPP has been developed, which includes the following methods: method for calculating TPP as a function of time by temperature inside the sample; TPP calculation method as a function of time by temperature on sample surfaces; TPP calculation method as a function of temperature, based on the integral transformation of the inverse heat conduction problem in the form of solving an integrofunctional equation for temperatures inside the sample; TPP calculation method as a function of time based on Kalman filter.
A generalized structure of supporting DMD subsystems for studying the properties of PCs, their components and the relationships between them, which implements experimental support, is built and presented. The construction of DMD software is considered. A method of simulation study of accuracy and evaluation of the possibility of using algorithms for calculating TPP is proposed. A subsystem for choosing the optimal method and algorithm for calculating TPP of polymer composites is proposed. The subsystem is built on the alternative use of four TPP calculation algorithms. The use of four algorithms has an undoubted advantage, since they complement each other's capabilities and expand the range of the conditions for performing experiments. The proposed decision support module for choosing the optimal TPP PC calculation algorithm reduces the error in determining TPP. An additional advantage of using a subsystem in DMD is the information support for users who do not have special knowledge about the principles of building and operating of methods for determining TPP as well as calculation algorithms and optimal conditions for their use, which expands the capabilities of the DMD user interface.
All developed algorithms for calculating TPP in DMD and descriptions of their use in DSM are presented by separate independent procedures. Such an architecture for building DMD software makes it possible in the future to increase the number of algorithms, thereby increasing the accuracy of calculating TPP and expanding the functionality of DMD.