Numerical Method
The variational approach applied to the full functional of the respective nonlinear boundary problem is used to obtain governing equations of the shell equilibrium in a nonlinear case. The Lagrange multipliers transformation is used to include boundary co
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Numerical Method
Abstract The variational approach applied to the full functional of the respective nonlinear boundary problem is used to obtain governing equations of the shell equilibrium in a nonlinear case. The Lagrange multipliers transformation is used to include boundary conditions into the system functional. The two-dimensional boundary problem is reduced to the sequence of one-dimensional ones by a separation of variables; proper integral coefficients of its linear and nonlinear terms are derived. The respective iterative process is equivalent to the extended Kantarovich method applied to a nonlinear boundary problem. The correspondence of the one-dimensional boundary problem and the respective Cauchy problem is demonstrated rendering the possibility of applying an efficient numerical technique to solve the latter. For determination of the equivalent initial vector of the Cauchy problem, the Newton method is used based on a Frechet derivative approximation calculation. The solution branches are traced with employment of the parameter continuation method using the natural—load—parameter or any suitable component of the initial vector. Finally, the complete numerical algorithm for building postcritical nonlinear solution branches is presented.
4.1 Governing Equations To solve nonlinear boundary problem (2.11–2.25) we apply a generalized solution approach. Vorovich (1999) proved that the respective equations are equivalent to those obtained from the variational approach with functional (2.27). Moreover, it was shown (Vorovich 1999) that to build a sequence minimizing the functional (for example, with employment of the iterative process considered below) is equivalent to finding the generalized solution to the correspondent boundary problem.
N. I. Obodan et al., Nonlinear Behaviour and Stability of Thin-Walled Shells, Solid Mechanics and Its Applications 199, DOI: 10.1007/978-94-007-6365-4_4, Springer Science+Business Media Dordrecht 2013
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Numerical Method
The solution of boundary problem (2.11), (2.12) may be presented as n o n o ðiÞ ði Þ ði Þ ; U ðx1 ; x2 Þ ¼ Uj ðx1 ; x2 Þ ¼ hj 1 ðx1 Þgj 2 ðx2 Þ; i ! 1 j¼1;12
ði Þ
ði Þ
hj 1 ðx1 Þgj 2 ðx2 Þ 2 Wtj
ð4:1Þ
where i1 ¼ i; i2 ¼ i 1; for even i; i1 ¼ i 1; i2 ¼ i; for odd i: Single-variable ðiÞ ðiÞ functions hj ðx1 Þ; gj ðx2 Þ are considered to be independently varied inside X: Thus, the respective equivalent variational formulation is 2 3 ð i Þ U ðx1 ; x2 Þ ¼ lim 4 arg min I Uj ðx1 ; x2 Þ 5 ð4:2Þ i!1
ði Þ
Uj ðx1 ;x2 Þ 2 Wtj
In the composition of the variational derivative of functional (2.27) with respect ðiÞ ð iÞ ði1Þ ði1Þ to functions hj ðx1 Þ and gj ðx2 Þ; the functions gj ðx2 Þ and hj ðx1 Þ, respectively, are considered to be known as they were found in the previous step of the iterative process. Let us point out that the iterative process determined by (4.2) is equivalent to the extended Kantorovich method (EKM) (Kantorovich and Krylov 1958; Kerr 1968; Aghdam et al. 2007) applied to solve nonlinear boundary problem (2.11)
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