Rayleigh-Ritz method

Klahn, B. and Bingel, W. A. [19 77] The Convergence of the Rayleigh-Ritz Method in Quantum Chemistry , Theoretica Chimica Acta, 44, p. 9. 

B. Klahn, W.A. Bingel, The convergence of the Rayleigh-Ritz method in quantum chemistry. I. The criteria for convergence, Theor. Chim. Acta 44 (1977) 9-26. 

On the other hand, Marin and Cruz [16-18] used the direct variational method or Rayleigh-Ritz method with a trial function of the same form used by Gorecki and Byers Brown (Equation (25)). Marin and Cruz chose hydrogenic functions with a variational parameter as the exponent for the functions

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Varshni [22] studied the CHA by means of the Rayleigh-Ritz method. He proposed a modification of wave function (26), introducing an additional variational parameter /9. In his approach, the Is, 2p and 3d CHA wave functions are written as... 

At the beginning of this section we presented a non-rigorous version of the Rayleigh-Ritz method for Dirac Hamiltonians. It is now time to examine the conditions under which this can be justified rigorously as a foundation for practical calculations with the DHF(B) equations. Our presentation here is based on [76,86,87] which give much more detail. 

Once the performance of the interface bond was established by the previous delamination (durability) tests, contoured double cantilever beam (CDCB) specimens were designed to conduct mode-I fracture tests. In this study, bilayer CDCB specimens (see Fig. 6) were designed by the Rayleigh-Ritz method [8] and used for fracture toughness tests on bonded FRP wood interfaces under both dry and wet conditions. The critical strain energy release rate, G/c, which is a measure of the fracture toughness, is given as ... 

Compute the compliance of the CDCB specimen by the Rayleigh-Ritz method [8] as a function of crack length and specimen-slope C = C(a,p). 

Comparison of Eqs. (h) and (q) with the exact solutions, Eq. (r), indicates that Rayleigh-Ritz method using two force-dependent Ritz vetors gives a roughly similar accuracy as modal analysis with two modes for responses 2> 3 U4, and us. However, the error in uj determined by modal analysis is large. This becomes obvious from the following table ... 

This procedure is referred to as the Rayleigh-Ritz method, which is equivalent to the Galerkin method. 

The critical buckling moments obtained from the solutions based on the developed formulation and the finite element model are shown in Table 1,2 for doubly symmetric section and in Table 3 for mono-symmetric section. In these tables, a comparison is made with the solutions proposed by Zhang Tong (2008) where the Rayleigh-Ritz method was used by the authors to solve this stability problem, as well as the critical loads results presented by Asgarian et al. (2013) in which the authors employed the finite element method by means of Ansys software and the numerical model by means of power series method. 

Variational principles play an important role in the solution of the Schrbdinger equation, e.g. the Ritz-method [5] for the solution of partial differential equations and the Rayleigh-Ritz method [6] for the calculation of bound states of atoms and molecules. The oldest variational method for scattering problems was introduced 1944 by Hulthen [7]. Four years later Hulthen and Kohn [8, 9, 10] independently developed what is now known as the Hulthen-Kohn... 

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