Fatigue cracks in machine components are subjected to stress fields induced by the external load and residual stresses resulting from the surface treatment. Stress fields in such cases are characterized by non-uniform distributions and handbook stress intensity factor solutions for such configurations are not available. The method presented below is based on the generalized weight function technique enabling the stress intensity factors to be calculated for any Mode I loading applied to arbitrary planar convex crack. The method is particularly suitable for modeling fatigue crack growth in presence of complex stress fields.

The accuracy and reliability of Kalman filter are easily affected by the gross errors in observations. Although robust Kalman filter based on equivalent weight function models can reduce the impact of gross errors on filtering results, the conventional equivalent weight function models are more suitable for the observations with the same noise level. For Precise Point Positioning (PPP) with multiple types of observations that have different measuring accuracy and noise levels, the filtering results obtained with conventional robust equivalent weight function models are not the best ones. For this problem, a classification robust equivalent weight function model based on the t-inspection statistics is proposed, which has better performance than the conventional equivalent weight function models in the case of no more than one gross error in a certain type of observations. However, in the case of multiple gross errors in a certain type of observations, the performance of the conventional robust Kalman filter based on the two kinds of equivalent weight function models are barely satisfactory due to the interaction between gross errors. To address this problem, an improved classification robust Kalman filtering method is further proposed in this paper. To verify and evaluate the performance of the proposed method, simulation tests were carried out based on the GPS/BDS data and their results were compared with those obtained with the conventional robust Kalman filtering method. The results show that the improved classification robust Kalman filtering method can effectively reduce the impact of multiple gross errors on the positioning results and significantly improve the positioning accuracy and reliability of PPP.

The critical plane orientations determined with account for maximum value of energy density parameters and the weight function method were compared to experimental fatigue fracture plane orientations. Energy density parameters used in two multiaxial fatigue failure criteria, i.e. (i) criterion of the maximum normal strain energy density on the critical plane and (ii) criterion of the maximum shear strain energy density on the critical plane were employed. In the other method, the weight functions were formed on the basis of energy parameters. These two methods were verified by experimental tests of 1802A steel. The material was subjected to cyclic and random bending, torsion and combined bending with torsion with different coefficients of cross correlation between normal and shear stresses. The calculated results are satisfactory for both methods.