Journal of Engineering Mechanics ASCE, 130 (2004), 720-732.


Borek Patzak
Czech Technical University in Prague
Milan Jirįsek
Swiss Federal Institute of Technology
LSC - ENAC, EPFL, 1015 Lausanne, Switzerland


This paper presents an adaptive mesh refinement technique suitable for the resolution of highly localized damage in concrete and other quasibrittle materials. Objectivity of the description of softening is ensured by using regularized material models based on the concept of nonlocal averaging, which is applied to isotropic and anisotropic damage  formulations. The distributions of strain and internal variables produced by such regularized models are continuous, which facilitates the projection of information from one finite element mesh onto another. However, not all mapping algorithms  for the transfer of internal variables preserve the basic characteristics of the localized process zone. The paper evaluates and compares three  mapping algorithms, which are based on the closest-point transfer, least-square projection, and shape-function projection. It also briefly comments on other important components of a complete adaptive strategy, i.e., on the error indicator, refinement rules and mesh generator. The efficiency of the proposed strategy is illustrated by examples that treat straight as well as curved crack trajectories. The underlying material model is a nonlocal integral formulation of anisotropic damage based on the microplane concept.

Concluding Remarks

In this paper we have presented and discussed various aspects of mesh-adaptive techniques applied to nonlocal continuum models with softening. Intuitively it is clear that adaptive mesh refinement can provide a good balance between accuracy and speed of the simulation, but the components of an adaptive strategy must be selected with care. For instance, we have shown that certain transfer operators for the mapping of internal variables can lead to an artificial diffusion of the process zone. Another important factor is the element size in the damage process zone, which must be small enough to permit nonlocal interaction of each Gauss point with a sufficient number of its neighbors. But even on a fine mesh, the actual curved crack trajectory is not reproduced correctly if the material model does not properly take into account the anisotropic character of damage.

The complete paper can be downloaded in an electronic form: gzipped PostScript, gzipped PDF

EPFL / 22 January 2003 /