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Multiskalenmodellierung und erweiterte finite Elmente Analyse von Bruchprozessen in Keramik

Multiscale modeling and extended finite element analysis of fracture processes in ceramics

Led by:  P. Wriggers, S. Löhnert
Team:  C. Prange
Year:  2009

The extended finite element method (XFEM) enables the modelling and calculation of cracks independent of mesh topologies. Due to this advantage it has become the most widely used method for computations of fracture processes. In the vicinity of an existing macrocrack tip microcracks can nucleate. These microcracks can lead to crack shielding or crack amplification effects. Therefore, microcracks in the vicinity of a macrocrack front have to be considered during the computation. This motivates the use of multiscale methods like the multiscale projection method that can capture fine scale effects where necessary.

Within the microscale computation a recovery based error estimation technique for discretisation errors is applied. In combination with mesh adaptation techniques this leads to error controlled and accurate stress distributions in the microdomain. To get comparable results for different meshes a relative error distribution for each element is calculated. Every element with an error greater than a certain threshold value is refined. During this process incompatible 'hanging' nodes occur which need a special treatment.

Figure 1 shows the error distribution for a single scale computation with one straight macrocrack. The discretisation error distribution of the adaptively refined mesh (Figure 1.b) agrees with the discretisation error distribution of the uniformly refined mesh (Figure 1.c).

      a) 2,800 DOF                              b) 150,000 DOF                     c) 2,800,000 DOF

 Figure 1: Relative error distribution for different meshes

During a multiscale computation the microscale domain has to be defined. Figure 2 shows the influence of the fine scale domain size. If this domain is not large enough (Figure 2.a) stress fluctuations at the boundary of the microdomain occur. For a larger fine scale domain these stress fluctuations decrease. In general a simple circular tube around the macrocrack front is not the optimal shape. Therefore a model error estimation is preferable to define the size and the shape of the microdomain independent of the macromesh descretisation.



a) 9 macro elements                  b) 21 macro elements         c) 45 macro elements

Figure 2: Influence of the radius of the fine scale domain