23 It should be noted that each iteration involves three computationally expensive steps:
1. Evaluation of internal forces fmt (or reactions)
2. Evaluation of the global tangent stiffness matrix Kt
3. Solution of a system of linear equations
24 This method is essentially the same as the Newton-Raphson however in Eq. 16.23 (K^) is replaced by Kt which is the tangent stiffness matrix of the first iteration of either 1) the first increment Kt = Ky0, Fig. 16.4, or 2) current increment, Fig. 16.3 Kt = Kyn Fig. 16.3
25 In general the cpu time required for the extra iterations required by this method is less than the one saved by the assembly and decomposition of the stiffness matrix for each iteration.
26 It should be mentioned that the tangent stiffness matrix does not necessarily have to be the true tangent stiffness matrix; an approximation of the true tangent stiffness matrix or even the initial stiffness matrix will generally produce satisfactory results, albeit at the cost of additional iterations.
27 This method is a compromise between the first two. First we seek two displacements by two cycles of modified Newton-Raphson, then a secant to the curve is established between those
two points, and a step taken along it, Fig. 16.5.
28 Subsequently, each step will be taken along a secant connecting the previous two points. Hence, starting with
the secant slope can be determined and then
29 This process can be generalized to (KS)
16.2.2 Acceleration of Convergence, Line Search Method Adapted from (Reich 1993)
30 The line search is an iterative technique for automatically under- or over-relaxing the displacement corrections so as to accelerate the convergence of nonlinear solution algorithms.
3i The amount of under- or over-relaxation is determined by enforcing an orthogonality condi-t.ion between the displacement, corrections and the residual loads TZ , which amounts to forcing the iterative change in energy to be zero.
32 The displacement corrections are multiplied by a scalar value sk defining the amount of under- or over-relaxation such that, the total displacements W+l,k are defined as
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