Finite Element Method - The Basis (Volume 1)

16 Some preliminaries: the standard discrete systemIn many complex problems an external constraint of some kind may be imagined,enforcing the requirement (1.25) with the number of degrees of freedom of a and a 0being quite di€erent. Even in such instances the relations (1.26) and (1.27) continueto be valid.An alternative and more general argument can be applied to many other situationsof discrete analysis. We wish to replace a set of parameters a in which the systemequations have been written by another one related to it by a transformationmatrix T asa ˆ Tb…1:30†In the linear case the system equations are of the formKa ˆ r ÿ f…1:31†and on the substitution we haveKTb ˆ r ÿ f…1:32†The new system can be premultiplied simply by T T , yielding…T T KT†b ˆ T T r ÿ T T f…1:33†which will preserve the symmetry of equations if the matrix K is symmetric. However,occasionally the matrix T is not square and expression (1.30) represents in fact anapproximation in which a larger number of parameters a is constrained. Clearly thesystem of equations (1.32) gives more equations than are necessary for a solutionof the reduced set of parameters b, and the ®nal expression (1.33) presents a reducedsystem which in some sense approximates the original one.We have thus introduced the basic idea of approximation, which will be the subjectof subsequent chapters where in®nite sets of quantities are reduced to ®nite sets.A historical development of the subject of ®nite element methods has been presentedby the author. 27;28References1. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, 1946.2. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, 1955.3. S.H. Crandall. Engineering Analysis. McGraw-Hill, 1956.4. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. AcademicPress, 1972.5. D. McHenry. A lattice analogy for the solution of plane stress problems. J. Inst. Civ. Eng.,21, 59±82, 1943.6. A. Hreniko€. Solution of problems in elasticity by the framework method. J. Appl. Mech.,A8, 169±75, 1941.7. N.M. Newmark. Numerical methods of analysis in bars, plates and elastic bodies, inNumerical Methods in Analysis in Engineering (ed. L.E. Grinter), Macmillan, 1949.8. J.H. Argyris. Energy Theorems and Structural Analysis. Butterworth, 1960 (reprinted fromAircraft Eng., 1954±5).9. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Sti€ness and de¯ection analysisof complex structures. J. Aero. Sci., 23, 805±23, 1956.