Transcription of Chapter 3 Classical Variational Methods and the Finite ...
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Chapter 3 Classical Variational Methods and the Finite ...
44 Chapter 3: Classical Variational Methods and the Finite Element Method Introduction Deriving the governing dynamics of physical processes is a complicated task in itself; finding exact solutions to the governing partial differential equations is usually even more formidable. When trying to solve such equations, approximate Methods of analysis provide a convenient, alternative method for finding solutions. Two such Methods , the Rayleigh-Ritz method and the Galerkin method, are typically used in the literature and are referred to as Classical Variational Methods . According to Reddy (1993), when solving a differential equation by a Variational method, the equation is first put into a weighted-integral form, and then the approximate solution within the domain of interest is assumed to be a linear combination )( iiic of appropriately chosen approximation functions i and undetermined coefficients, ci.
finite element method. 3.3.1 The Rayleigh-Ritz Method Before delving into the Rayleigh-Ritz method, a short historical perspective (summarized from Meirovitch (1997)) is in order. The method was first used by Lord Rayleigh in 1870 (Gould, 1995) to solve the vibration problem of organ pipes closed on one end and open at the other.
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