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CivilComp Proceedings
ISSN 17593433 CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 165
The Influence of NonLinear Material Properties on the Global Convergence in the FiniteElement Analysis of Spatial Beams D. Zupan and M. Saje
Faculty of Civil and Geodetic Engineering, University of Ljubljana, Slovenia D. Zupan, M. Saje, "The Influence of NonLinear Material Properties on the Global Convergence in the FiniteElement Analysis of Spatial Beams", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 165, 2005. doi:10.4203/ccp.81.165
Keywords: nonlinear beam theory, nonlinear material, reinforced concrete, stress field integration, Newton's iteration, rate of convergence.
Summary
In the finiteelement analysis of geometrically nonlinear spatial beams and
frame structures made of nonlinear material, the nonlinearity of the
constitutive law plays an important role in the rate of convergence of the
global iteration procedure. In this paper we study the influence of solving
the nonlinear constitutive equations in the global or local manner.
The discretized governing equations of the nonlinear spatial beams must be solved iteratively. In the finiteelement formulation, the choice of the primary unknowns seems to be crucial for the efficiency of the overall numerical algorithm. The constitutive equations can be eliminated from the set of the governing equations of the beam. This reduces the size of the global system of linearized equations that needs to be solved in each iteration. When the nonlinear material is taken into account, such a formulation must be capable of solving locally the constitutive equations in each step of the global iteration. Nonlinear constitutive equations must be solved iteratively. This implies that, in each step of the global iteration, several local iterations in integration points need to be executed. One would expect that such an approach would not dramatically increase the number of global iterations compared to linear material; this turns out to be true, yet the computational times are often substantially bigger. If the constitutive equations are taken to be the part of the global governing equations, the size of the global linearized system of equations is much larger, but the local iterations are not needed. Such an approach may somewhat increase the number of global iterations, but the overall computational cost might be lower. When solving the governing equations of the beam, the constitutive equations can be treated in several ways. Three different approaches will be analyzed in the paper:
A short outline of each approach will be presented. We will also briefly describe three different numerical methods which will be used for comparisons of the convergence rate and the overall computational cost of the above approaches. In order to make the comparisons unbiased, the computer code for all of the formulations shares the same pre and postprocessing algorithms and all of the common parts are programed identically. Computer programs were developed and tested using Matlab [1], based on the algorithms, presented in [2,3]. We used the same type of the discretization in all formulations. The influence of different approaches on the rate of convergence and the overall computational costs will be examined through several numerical examples. For each particular case we will compare not only the number of iterations but also the total number of floating point operations. The total number of floating point operations reveals the actual computational cost of the algorithm independently on the computer equipment used for the studies. We have chosen to compare the three formulations on realistic problems using realistic material model. The data were obtained from well documented experimental tests of reinforced concrete beams and frames. As the differences in the computational costs between the approaches grow if the applied loads approach the critical ones, all the tests are performed in the neighbourhood of the critical loading. References
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