Computational model boundary conditions

Industrial scale polymer forming operations are usually based on the combination of various types of individual processes. Therefore in the computer-aided design of these operations a section-by-section approach can be adopted, in which each section of a larger process is modelled separately. An important requirement in this approach is the imposition of realistic boundary conditions at the limits of the sub-sections of a complicated process. The division of a complex operation into simpler sections should therefore be based on a systematic procedure that can provide the necessary boundary conditions at the limits of its sub-processes. A rational method for the identification of the subprocesses of common types of polymer forming operations is described by Tadmor and Gogos (1979). 

There are cases where non-regular lattices may be of advantage [36,37]. The computational effort, however, is substantially larger, which makes the models less flexible concerning changes of boundary conditions or topological constraints. Another direction, which may be promising in the future, is the use of hybrid models, where for example local attachment kinetics are treated on a microscopic atomistic scale, while the transport properties are treated by macroscopic partial differential equations [5,6]. 

The first step in applying FEA is the construction of a model that breaks a component into simple standardized shapes or (usual term) elements located in space by a common coordinate grid system. The coordinate points of the element corners, or nodes, are the locations in the model where output data are provided. In some cases, special elements can also be used that provide additional nodes along their length or sides. Nodal stiffness properties are identified, arranged into matrices, and loaded into a computer where they are processed with certain applied loads and boundary conditions to calculate displacements and strains imposed by the loads (Appendix A PLASTICS DESIGN TOOLBOX). 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

The program can solve both steady-state problems as well as time-dependent problems, and has provisions for both linear and nonlinear problems. The boundary conditions and material properties can vary with time, temperature, and position. The property variation with position can be a straight line function or or a series of connected straight line functions. User-written Fortran subroutines can be used to implement more exotic changes of boundary conditions, material properties, or to model control systems. The program has been implemented on MS DOS microcomputers, VAX computers, and CRAY supercomputers. The present work used the MS DOS microcomputer implementation. 

An Eulerian-Eulerian (EE) approach was adopted to simulate the dispersed gas-liquid flow. The EE approach treats both the primary liquid phase and the dispersed gas phase as interpenetrating continua, and solves a set of Navier-Stokes equations for each phase. Velocity inlet and outlet boundary conditions were employed in the liquid phase, whilst the gas phase conditions consisted of a velocity inlet and pressure outlet. Turbulence within the system was account for with the Standard k-e model, implemented on a per-phase basis, similar to the recent work of Bertola et. al.[4]. A more detailed description of the computational setup of the EE method can be found in Pareek et. al.[5]. 

In order to describe correctly the dynamic evolution of a fluid/fluid interface, a number of boundary conditions have to be implemented into the computational models. 

Further progress in understanding membrane instability and nonlocality requires development of microscopic theory and modeling. Analysis of membrane thickness fluctuations derived from molecular dynamics simulations can serve such a purpose. A possible difficulty with such analysis must be mentioned. In a natural environment isolated membranes assume a stressless state. However, MD modeling requires imposition of special boundary conditions corresponding to a stressed state of the membrane (see Refs. 84,87,112). This stress can interfere with the fluctuations of membrane shape and thickness, an effect that must be accounted for in analyzing data extracted from computer experiments. 

The modeling of complex solids has greatly advanced since the advent, around 1960, of the finite element method [196], Here the material is divided into a number of subdomains, termed elements, with associated nodes. The elements are considered to consist of materials, the constitutive equations of which are well known, and, upon change of the system, the nodes suffer nodal displacements and concomitant generalized nodal forces. The method involves construction of a global stiffness matrix that comprises the contributions from all elements, the relevant boundary conditions and body and thermal forces a typical problem is then to compute the nodal displacements (i. e., the local strains) by solving the system K u = F, where K is the stiffness matrix, u the... 

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