Page 364 - Process Modelling and Simulation With Finite Element Methods

Appendix

A MATLABmEMLAB PRIMER FOR VECTOR CALCULUS

W.B.J. ZIMMEFWAN
Department of Chemical and Process Engineering, University of Shefield,
Newcastle Street, Shefield SI 3JD United Kingdom

and

J.M. REES
Department of Applied Mathematics, University of Shefield, Hicks Building, Sheffield

Vector calculus underpins partial differential equations and their numerical
approximation. Modelers must have a good working knowledge of the basics of vector
calculus to use finite element methods effectively. Perhaps because undergraduate
engineers are not confronted with realistic applications of vector calculus, but rather
learn it as a mathematical discipline, their ability to apply vector calculus in engineering
modeling is limited. In this appendix, all the basics of vector calculus are introduced
with reference to MATLAB/FEMLAB utility and implementation. So the other way of
reading this appendix is as a primer for MATLAEWEMLAB basics with regard to
multivariable differential calculus. When we wrote this appendix, we debated whether or
not to augment Chapter One (basics of numerical analysis) with the material directly, as
numerical approximation of derivatives is fundamental to the solution of PDEs - a
FEMLAB “primitive” operation. Indeed, in learning spectral methods for solving PDEs,
the fundamental theorem is the “derivative theorem” - how to use the spectral transform
method to approximate derivatives. So by analogy, the fundamental utility of FEM is
numerical differentiation. The debate was lost in that Chapter One aims to solve basic
problems with FEMLAB straightaway. Approximating derivatives, no matter how
useful, is still an intermediate step in modeling, rarely the objective itself. The only
MATLAB basics we consider essential that are not used in making vector calculus the
point of this appendix are eigenvalue analysis and logical expressions. These are
sprinkled throughout the textbook anyway.

A.1 Review of Vectors
A. 1.1 Representation of vectors

Since FEMLAB deals with scalar, vector, and matrix quantities, if only as input
coefficients, a brief review of the representation of vectors (as a special case of
MATLAB’s matrix data type) is in order here. Scalar quantities can be
represented by a single number, but vector quantities have magnitude and
direction. Given a righthanded coordinate system as shown in Figure Al, any
vector a is expressible in the form

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