[理学]1The finite element method-its____ basis and fundamentals.ppt

1.The standard discrete system and origins of the finite element method 1.8 Transformation of coordinates An alternative and more general argument can be applied to many other situations of discrete analysis. We wish to replace a set of parameters u in which the system equations have been written by another one related to it by a transformation matrix T as In the linear case the system equations are of the form and on the substitution we have The new system can be premultiplied simply by TT , yielding 1.The standard discrete system and origins of the finite element method 1.8 Transformation of coordinates which will preserve the symmetry of equations if the matrix K is symmetric. However, occasionally the matrix T is not square and expression (1.26) represents in fact an approximation in which a larger number of parameters u is constrained. Clearly the system of equations (1.28) gives more equations than are necessary for a solution of the reduced set of parameters v, and the final expression (1.29) presents a reduced system which in some sense approximates the original one. We have thus introduced the basic idea of approximation, which will be the subject of subsequent chapters where infinite sets of quantities are reduced to finite sets. 1.The standard discrete system and origins of the finite element method 1.9 Problems 1.1 A simple fluid network to transport water is shown in Fig. 1.4. Each ‘element’ of the network is modelled in terms of the flow, J, and head, V, which are approximated by the linear relation where Ke is the coefficient array for element (e). The individual terms in the flow vector denote the total amount of flow entering ( + ) or leaving ( ? ) each end point. The properties of the elements are given by for elements 1 and 4, and for elements 2 and 3 by 1.The standard discrete system and origins of the finite element method 1.9 Problems where ce is an element related parameter. The system is operatin