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The application of the described h-adaptive methodology is presented on a 2D simulation of the brazilian splitting test, which is a standard technique for determination of the tensile strength of concrete. In this test, a cylindrical specimen is loaded along its vertical diametral plane. The compressive load, transfered to the specimen via steel bearing plates at the top and bottom sides, induces tension stress in the horizontal direction leading finally to the rupture of the specimen along the loading plane. Due to the double symmetry, the analysis itself is performed only on the quarter of the specimen under plain strain conditions. The concrete behaviour is described by the nonlocal scalar damage model, while the steel bearing plates are assumed to be linearly elastic. The considered dimensions of the specimen and the relevant material parameters are shown in Fig. 3 on the left.

The analysis was initially performed without the adaptive refinement using the discretization depicted in Fig. 3 on the right. The nonlinear problem was solved incrementally applying the cylindrical arc length method taking the horizontal displacement of point B (see Fig. 3) as the arc length control parameter. The highly nonlinear response of the specimen is displayed in terms of the load-displacement diagrams of points A and B in Fig. 4. Note the severe snap-back and the slight reloading on the loading path of point A, which is in qualitative agreement with the experimental observations [23]. The obtained results are illustrated in Fig. 5, where the damage profiles are plotted at states I, II, III and IV marked in Fig. 4. The damage initiates at the centre of the specimen (state I) and propagates along the loading plane (state II) until it reaches the bearing plates (state III). At this stage the specimen is effectively split to two parts that work further separately under compression until the wedge of damage is formed under the steel bearing plates (state IV). The failure of the specimen at this stage observed in experiments is not captured by the simulation due to the adopted scalar damage model tuned to unload the specimen thoroughly. Note that structural response of the brazilian splitting test is sensitive to the actual arrangement of the test. Generally, the mechanism of failure is affected by various factors, especially the ratio of the specimen radius to the bearing plate width [23].

To assess quantitatively the quality of the obtained solution the error estimation has been carried on at each of the 63 steps of the nonlinear analysis. The resulting profile of the evolution of the relative error is displayed in Fig. 6 together with the marks corresponding to the states discussed above. The relative error, initially at reasonable level of 8 %, exceeds the acceptable value of 10 % approximately at state II, then dramatically increases to maximal value of 27 % near state III, drops down to 5 % at state IV and grows up again up to 12 % at the end of the analysis. Thus the reason to perform an adaptive analysis keeping the error below a prescribed threshold is obvious.

The adaptive simulation of the brazilian splitting test was performed with the target relative error 10 % (taken as a common engineering tolerance and being similar to the initial relative error of the simulation accomplished without adaptive refinement). The evolution of the relative error, controlled by the adaptive analysis, is sketched in Fig. 7. The mesh used for the nonadaptive computation was employed as the initial mesh for the adaptivity. This mesh was simultaneously used as the coarsest allowable mesh in order to prevent derefinement of the localization zone (especially at the late stages of the analysis) to ensure the nonlocal model is working properly. From the error evolution it is also clear that the global relative error defined by Eq. (16) was not used to trigger the rediscretization of the domain. Instead, the weighted global relative error that attempts to account for the nonuniform error distribution was employed. It is considered as




The evolution of the weighted error is depicted in Fig. 8 which reveals that decrease of the weighted global relative error bellow 2.5 % was used to initiate the derefinement. The arc length procedure was controlled by the same parameter as in the nonadaptive simulation. The obtained structural response in terms of loading paths of points A and B is depicted in Fig. 9. The individual curves correspond to the response evaluated for consequent discretizations that are summarized in Figure 10. The overlap of subsequent pairs of curves in Figs 7, 8 and 9 is related to the restart of the analysis from the previous step. Surprisingly, there is not much difference in the response (given by the loading paths of points A and B) between the adaptive and nonadaptive analysis. This verifies the objectivity of the applied nonlocal continuum approach giving the same global response (note that points A and B are actually outside of the fracture process zone) independent of the discretization, provided that the elements in the localization zone are smaller than the material characteristic length. The only significant difference at state III can be attributed to the mapping error. The benefit of the adaptive solution consists in the higher resolution of the localization zone resulting in strain profile captured with higher accuracy as it is shown in Fig. 11 at analysis step 33. There is effectively no difference in the damage distribution, which is depicted for the same step in Fig. 12. It is also evident that the mesh refinement (based on the estimated error) follows rather the strain gradient than the damage gradient which was reported in [24]. This can be illustrated on error distribution again at the step 33 for consequent meshes 3 and 4 in Fig. 13.

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Daniel Rypl