Finite Element Method - The Basis (Volume 1)

14 Some preliminaries: the standard discrete systemsubject of equation solving, though extremely important, is in general beyond thescope of this book.The ®nal step discussed above can be followed by substitution to obtain stresses,currents, or other desired output quantities.All operations involved in structural or other network analysis are thus of anextremely simple and repetitive kind.We can now de®ne the standard discrete system as one in which such conditionsprevail.1.7 The standard discrete systemIn the standard discrete system, whether it is structural or of any other kind, we ®ndthat:1. A set of discrete parameters, say a i , can be identi®ed which describes simultaneouslythe behaviour of each element, e, and of the whole system. We shall callthese the system parameters.2. For each element a set of quantities q e i can be computed in terms of the systemparameters a i . The general function relationship can be non-linearq e i ˆ q e i …a†…1:20†but in many cases a linear form exists givingq e i ˆ K e i 1a 1 ‡ K e i 2a 2 ‡‡f e i…1:21†3. The system equations are obtained by a simple additionr i ˆ Xm…1:22†q e ie ˆ 1where r i are system quantities (often prescribed as zero).In the linear case this results in a system of equationsKa ‡ f ˆ rsuch thatK ij ˆ Xmf i ˆ XmK e ije ˆ 1f e ie ˆ 1…1:23†…1:24†from which the solution for the system variables a can be found after imposingnecessary boundary conditions.The reader will observe that this de®nition includes the structural, hydraulic, andelectrical examples already discussed. However, it is broader. In general neitherlinearity nor symmetry of matrices need exist ± although in many problems thiswill arise naturally. Further, the narrowness of interconnections existing in usualelements is not essential.While much further detail could be discussed (we refer the reader to speci®c booksfor more exhaustive studies in the structural context 24ÿ26 ), we feel that the generalexpose given here should su ce for further study of this book.