Analysis of Calendering Using FEM

The FEM, which was originally developed for structural analysis of solids, has been very successfully applied in the past decades to viscous fluid flow as well. In fact, with the exponentially growing computer power, it has become a practical and indispensable tool for solving complex viscous and viscoelastic flows in polymer processing (20) and it is the core of the quickly developing discipline of computational fluid mechanics (cf. Section 7.5). 

One of the first applications of FEM in polymer processing is a result of the work of Vlachopoulos and Kiparissides (21,22). Some of the computed results obtained by this 

The principles and applications of FEM are described extensively in the literature (e.g., 23-26). FEM is a numerical approximation to continuum problems that provides an approximate, piecewise, continuous representation of the unknown field variables (e.g., pressures, velocities). 

The continuous region or body is subdivided into a finite number of subregions or elements (Fig. 15.5). The elements may be of variable size and shape, and they are so chosen because they closely fit the body. This is in sharp contrast to finite difference methods, which are characterized by a regular size mesh, describable by the coordinates that describe the boundaries of the body. 

The crossing of two curves bounding adjacent elements form nodes. The values of the field variables at the nodes form the desired solution. Common shapes of finite elements are triangular, rectangular, and quadrilateral in two-dimensional problems, and rectangular, prismatic, and tetrahedral in three-dimensional problems. Within each element, an interpolation function for the variable is assumed. These assumed functions, called trial functions or field variable models, are relatively simple functions such as truncated polynomials. The number of terms (coefficients) in the polynomial selected to represent the unknown function must at least equal the degrees of freedom associated with the element. For example, in a simple one-dimensional case [Fig. 15.6(a)], we have two degrees of freedom, Pt and Pj, for a field variable P(x) in element e. Additional conditions are needed for more terms (e.g., derivatives at nodes i and j or additional internal nodes). 

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