Mathematical model implementation methods

The mathematical model implemented is based on differential equations of heat and mass transfer along the time and they are numerically solved using the Runge-Kutta-Merson method. The final program is implemented in Fortran. Different time steps delta t = 0.001 s and 0.0001s) were used for the numerical integration and the same results were obtained. The objective of the software is to estimate temperatures and humidity in the different parts of the body, as function of time (Teixeira et al., 2010). 

Solving the nonlinear mathematic models by a numerical method depends on the trial values that are chosen by experience and is not based upon a particular theory. The results obtained are greatly influenced by the initial values selected that is a characteristic of strong nonlinear problems. By using numerical techniques to solve nonlinear models, iterations must be implemented. With the incorrect selection of a trial value, divergence in solution can appear. The accuracy of the numerical results cannot be estimated theoretically for nonlinear problem since there is no analytical solution available. This is why an approximate analytical solution is extremely useful for a theoretical analysis of nonlinear problem If an approximate analytical solution can be obtained, then this has a number of benefits ... 

Another approach to the data based on low-level counting uses the method of maximum likelihood. The likelihood of a set of data is the probability of obtaining the particular set, given the chosen probability distribution model. The idea is to determine the parameters that maximize the likelihood of the sample data. The methodology is simple, but the implementation may need intense mathematics [12], The method has been used, for instance, to treat data on production rates [12] and... 

Reusing parts of VeDa and of the VeDa-based implementation models of the afore mentioned research prototypes ROME and ModKit (cf. Sect. 5.3), the partial model Mathematical Models comprises both generic concepts for mathematical modeling and specific types of mathematical models. Within the partial model Unstructured Models, models are described from a mathematical point of view. Concepts of Unstructured Models can be used to specify the equations of System Models, which model Systems in a structured manner. Models for the description of ChemicalProcessSystems (as defined in CPS Models) are examples for system models. Such models employ concepts from Material Model to describe the behavior of the materials processed by the chemical process system. Finally, the partial model Cost Models introduces methods to estimate the cost for construction, procurement, and operation of chemical process systems. 

The physical context concept in the conceptual model is extended to describe the behavior of plastics in the form of pellets through the class solid state condition which encapsulates properties such as pellet type. This part of the implementation model concerns the mathematical modeling of some of the properties of polymers, which correspond to their djmamic or flow behavior. A class for a concrete mathematical model not only holds declarative information such as the list of parameters, but also provides a method for calculating the value of the property modeled. This method requires an implementation which is usually different from the one for another mathematical model. Therefore, mathematical models are organized in this application through further classification. 

The SIP methods by Stone (1986) for the gas leak flow equations and the rock mass deformation equations for double parallel coal seams have been developed by Sun (1998,2002) with Microsoft Visual C+-I- 6.0 under Windows2000 on a Pentium rv rc. The program is suitable for isotropic heterogeneous coal seams with irregular shapes as well as anisotropic heterogeneous coal seams. The systems with the first or the second boundary conditions as well as the hybrid boundary conditions can all be solved. The overview of the numerical implementations of the SIP methods for the solid-gas coupled mathematical model for double parallel coal seams is described as follows. 

The main advantage of the boundary integral methods, compared with other methods (e.g., FD, FE), is that for a number of multiphase flow problems, its implementation involves integration on the interfaces only. Thus, discretization is required only for the interfaces, which allows for higher accuracy and performance, especially in three-dimensional simulations. An important feature of the mathematical model used in the Bl method is that the velocity at a given time instant depends only on the position of the interfaces at that time instant. 

The study for predicting the velocity profiles in pultrusion by Gorthala et aL (1994) used a variable viscosity model. A comprehensive two-dimensional mathematical model in cylindrical coordinates was developed for resin flow, cure and heat transfer. A control-volume-based finite difference method (FDM) (Patankar method) was used for solving the governing equations. The use of artificial neural networks (ANNs) for pultrusion modelling in terms of the real process data and their potential for intelligent machine control was proposed by Wilcox and Wright (1998). Liu et al. (2000 Liu, 2001 Liu and Hillier, 1999) implemented a finite element/control volume... 

A general overview in Sect. 5.2 about the simulation of adaptronic (mechanical) systems is followed by a discussion of steps to be taken towards a mathematical model of an adaptronic structure in Sect. 5.3. Once a mathematical model of the adaptronic system has been derived and implemented numerically, analysis and simulations have to be carried out to characterise its dynamic behaviour. A survey of related methods and algorithms is given in Sect. 5.4. Simulation goals such as stability, performance and robustness are discussed, especially for the case of actively controlled structures. The modelling and simulation process is also demonstrated by a practical example in Sect. 5.5, while Sect. 5.6 gives an outlook on adaptronic system optimisation techniques. 

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