Keywords: adaptive analysis, error estimation, nonlinear analysis, strain softening, damage, nonlocal continuum.

The finite element method has become the most powerful tool for structural analysis. During the last decades, the method has matured to such a state that it is massively used in practical engineering for the solution of broad range of problems starting from linear elasticity up to highly non-linear transient simulations of the behaviour of real materials and complex structures. However, it is often applied without good understanding of the method's background, leading to incorrect results and inadequate design, possibly causing damage or failure of the structure. A good way to prevent these undesirable effects is to check the quality of the obtained solution. If the results do not meet the prescribed level of accuracy, the discretization of the problem must be adequately adjusted and the problem recalculated. This process of solution enhancement is called the adaptive analysis.

The failure of quasi-brittle materials is characterized by the development of fracture process zone. Modelling of progressive growth of microcracks and their coalescence in this zone leads to constitutive laws with strain softening. Standard continuum approaches to this phenomenon suffers from serious mathematical and numerical difficulties manifested by sensitivity to element size. The use of appropriate regularization technique ensuring mesh independent response and the objectivity of the numerical solution is therefore required. The present contribution deals with the regularization based on the nonlocal averaging technique which is recognized as a powerful localization limiter.

To ensure the quality of the finite element solution an adaptive strategy, keeping the error within the user defined limits, is applied. While there are several reliable methods for error assessment for linear problems [1,2], the error estimation for non-linear problems is still far from being matured. This is especially true for materials with strain softening. A recently published error estimation strategy by Huerta et.al. [3,4] seems to be quite promising. It is a residual based error estimator with the same methodology for the linear and nonlinear cases avoiding the demanding computation of equilibrating boundary fluxes. It enables to capture both the material as well as geometrical nonlinearity of the underlying problem and is well suited for a wide range of engineering problems.

This error estimator is used in the context of the isotropic damage model regularized by nonlocal continuum and is applied within a truly adaptive framework which combines properly the finite element solver for the considered problem, the error estimator, a suitable refinement strategy, a mesh generator capable to follow a prescribed mesh density distribution and a dispersion free mapping operator. After each step of the nonlinear analysis the solution error is estimated. If it exceeds the prescribed limit a new discretization, based on the refinement strategy taking into account the actual spatial error distribution, is created. The solution state (primary unknowns and state variables) is transferred to the new mesh using the mapping operator and the solution process continues on the new discretization by restarting the analysis from the previous step at which the error was still bellow the user prescribed threshold. This process is repeated until the analysis is finished.

The proposed methodology is presented on a 2D example demonstrating the vitality of the approach in terms of the obtained structural response. On the other hand it also reveals its large computational cost, which makes the analyses prohibitively expensive especially for large and complex 3D structures. A possible remedy consists in the parallelization of the analysis (including the error estimation) which is subject of further development.

1

O.C. Zienkiewicz, J.Z. Zhu, "A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis", International Journal for Numerical Methods in Engineering, 24, 337-357, 1987. doi:10.1002/nme.1620240206

2

O.C. Zienkiewicz, J.Z. Zhu, "The Superconvergent Patch Recovery (SPR) and Adaptive Finite Element Refinement", Computer Methods in Applied Mechanics in Engineering, 101, 207-224, 1992. doi:10.1016/0045-7825(92)90023-D

3

P. Diez, J.J. Egozcue, A. Huerta, "A Posteriori Error Estimation for Standard Finite Element Analysis", Computer Methods in Applied Mechanics in Engineering, 163, 141-157, 1998. doi:10.1016/S0045-7825(98)00009-7

4

A. Huerta, P. Diez, "Error Estimation Including Pollution Assessment for Nonlinear Finite Element Analysis", Computer Methods in Applied Mechanics in Engineering, 181, 21-41, 2000. doi:10.1016/S0045-7825(99)00071-7

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