Boundary value problems overview

Finite element methods are one of several approximate numerical techniques available for the solution of engineering boundary value problems. Analysis of materials processing operations lead to equations of this type, and finite element methods have a number of advantages in modeling such processes. This document is intended as an overview of this technique, to include examples relevant to polymer processing technology. 

A differential equation that has data given at more than one value of the independent variable is a boundary-value problem (BVP). Consequently, the differential equation must be of at least second order. The solution methods for BVPs are different compared to the methods used for initial-value problems (IVPs). An overview of a few of these methods will be presented in Sections 6.2.1. 2.3. The shooting method is the first method presented. It actually allows initial-value methods to be used, in that it transforms a BVP to an IVP, and finds the solution for the IVP. The lack of boundary conditions at the beginning of the interval requires several IVPs to be solved before the solution converges with the BVP solution. Another method presented later on is the finite difference method, which solves the BVP by converting the differential equation and the boundary conditions to a system of linear or non-hnear equations. Finally, the collocation and finite element methods, which solve the BVP by approximating the solution in terms of basis functions, are presented. 

The calculation of NMR parameter has been studied extensively see [3, 73] for general overviews. In 2001, Sebastian and Parrinello implemented the NMR chemical shift calculation in the plane wave AIMD code CPMD [74]. From this implementation it was possible to treat extended systems within periodic boundary conditions, i.e., the method was applicable to crystalline and amorphous insulators as well as to liquids. The problem of the position operator was solved by the use of maximally localized Wannier functions. Several benchmark calculations showed good agreement with experimental values. 

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