October AMP_Digital

A D V A N C E D M A T E R I A L S & P R O C E S S E S | O C T O B E R 2 0 1 8 2 2 Mathematical models can be used instead of physical experiments if the models are sufficiently reliable. Models show the effects of variables in different situations and can be used to improve product quality, make process chang- es, and determine ways to produce new products with desired property combi- nations. Because there are often sever- al ways to achieve the desired result, it is possible to minimize variables such as energy consumption and produc- tion cost while determining ways to at- tain the required material properties. In general, mathematical models can help speed up product, materials, and process development, and can also be used to estimate variables that are diffi- cult to measure. Various approaches can be used in mathematical modeling for differ- ent situations. Physical or phenomeno- logical modeling involves writing laws of nature in mathematical form to de- scribe a process or phenomenon. This usually requires many assumptions and simplifications and is not particularly effective for predicting material behav- ior. Empirical and semi-empirical mod- eling are based on observations and do not require assumptions or simpli- fications. Observations can stem from production or experimental data or a combination of the two. Such models describe reality as seen from the data. However, conventional techniques of empirical modeling are linear. NONLINEAR MODELING Nothing in nature is very linear, and materials science is also full of nonlinearities. Conventional methods of empirical modeling involve linear statistical techniques, which are inef- ficient at treating nonlinearities even when nonlinear terms are used. New methods of nonlinear modeling are based on free-form nonlinearities and do not require advance knowledge of the type of nonlinearities. Nonlinear modeling is empirical or semi-empirical modeling that takes into account some nonlinearities and can be carried out using various methods. The older tech- niques include polynomial regression, linear regression with nonlinear terms, and nonlinear regression. Newer methods include feed-for- ward neural networks, which do not require a priori knowledge of the nonlin- earities in the relationships. Among the new methods, feed-forward neural net- works are attractive due to their univer- sal approximation capabilities [1] , which make them suitable for most function approximation tasks that exist in mate- rials science and process engineering. Besides their universal approximation capability, it is usually possible to pro- duce nonlinear models with extrapo- lation capabilities. The user does not need to know the type and severity of nonlinearities while developing the models. Neural networks have been used for more than 20 years [2] for materials and process development in industries including plastics [3] , rubbers [4] , cements and concretes [5,6] , metals [7,8] , medical materials [9] , ceramics [10] , chemicals, pow- er generation, semiconductors, and biotechnology. VARIABLES OF FOAMED PLASTICS Depending on the application, cer- tain properties of foamed plastics are covered by specifications. These prop- erties depend on the amount of poly- mer, blowing agent, cross-linking agent, and other ingredients fed into the pro- cess, along with extrusion process vari- ables, further processing in the oven, and several dimensional variables. In the Furukawa process used by NMC Termonova, polyethylene is ex- truded with a blowing agent and a cross-linking agent as thick sheets and slabs, which pass through an oven where cross-linking and foaming oc- curs. Figure 3 shows a typical config- uration for nonlinear models of the process. Composition variables in- clude the amount of polymer, blowing Fig. 2 — Schematic of nonlinear models of material properties with production economics. Fig. 3 — Nonlinear models of cross-linking and foaming of polyethylene.