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The method of successive approximations, discussed by Engesser (1893) and Vianello (1898), was extensively employed in the “pre-FEM” literature. Sekhniashvili (1966) utilizes this method, as discussed by Bernshtein (1941) with attendant theoretical justification provided by Sushenkov (see Papkovich, 1933), who showed that in the limit the method leads to the fundamental frequency and the corresponding mode shape. Later, the method was utilized by Ratzensdorfer (1943), Popovich (1962), Mitelman (1970), Hodges (1997) and others.
Elaborated by Walter Ritz (1909) and the Rayleigh–Ritz method is now widely used not only in studying vibration but in solving problems in elasticity, theory of structures, nonlinear mechanics, and other branches of physics. Perhaps no other single mathematical tool has led to as much research in the strength of materials and theory of elasticity. © 2005 by Issac Elishakoff EOIS: “2892_chap01” — 2004/9/28 — 19:18 — page 36 — #30 Introduction 37 The first application of Rayleigh’s method to the buckling problem was performed by Bryan (1888).
If we start skiing from different places on the slope, and on each occasion we time our arrival at a fixed place on the valley floor, how can we calculate the topographic profile of undulations of the trail? This is the inverse problem. It is certainly a practical problem. It is also challenging and difficult and, indeed, in the general sense, it has no unique solution. According to Tarantola (1987), To solve the forward problem is to predict the values of the observable parameters, given arbitrary values of the model parameters.
