On Improving the Celebrated Paris’ Power Law for Fatigue,by Using Moving Least Squares

In this study, we propose to approximate the a－n relation as well as the da/dn－∆K relation, in fatigue crack propagation, by using the Moving Least Squares (MLS) method. This simple approach can avoid the internal inconsistencies caused by the celebrated Paris’ power law approximation of the da/dn－∆K relation, as well as the error caused by a simple numerical differentiation of the noisy data for a－n measurements in standard fatigue tests. Efficient, accurate and automatic simulations of fatigue crack propagation can, in general, be realized by using the currently developed MLS law as the “fatigue engine” [da/dn versus ∆K], and using a high-performance “fracture engine” [computing the K-factors] such as the Finite Element Alternating Method.
In the present paper, the “fatigue engine” based on the present MLS law, and the “fracture engine” based on the SafeFlaw computer program developed earlier by the authors, in conjunction with the COTS software ANSYS, were used for predicting the total life of arbitrarily cracked structures.
By comparing the numerical simulations with experimental tests, it is demonstrated that the current approach can give excellent predictions of the total fatigue life of a cracked structure, while the celebrated Paris’ Power Law may miscalculate the total fatigue life by a very large amount.     

On Condition Numbers for the Weighted Moore-Penrose Inverse and the Weighted Least Squares Problem involving Kronecker Products

We establish some explicit expressions for norm-wise, mixed and componentwise condition numbers for the weighted Moore-Penrose inverse of a matrix $A⊗B$ and more general matrix function compositions involving Kronecker products. The condition number for the weighted least squares problem (WLS) involving a Kronecker product is also discussed.     

How to search the experimental conditions that improve a Partial Least Squares calibration model: Application to a flow system with electrochemical detection for the determination of sulfonamides in milk

This paper deals with the selection of experimental conditions and how the signals obtained in these conditions influence the fitted Partial Least Squares calibration model. The multivariate signals come from a flow analysis system with amperometric detection when determining sulfadiazine, sulfamerazine and sulfamethazine in milk.The solution (carrier plus analyte) was pumped through the system to provide a continuous supply of analyte to the cell. The detector was programmed for a scan mode operation being the multivariate signal the hydrodynamic voltammogram. To obtain an analytical signal of enough analytical quality, the Net Analyte Signal and its standard deviation have been optimised by using an experimental design. The conflicting behaviour of the two responses has been solved by estimating the Pareto-optimal front.The multivariate signals recorded in the optimal conditions found have been calibrated by Partial Least Squares regression and their figures of merit validated according to the criteria established in European Decision 2002/657/EC.In relation to the permitted limit, 100 µg l^− 1 in milk, for the total content of sulfonamides established in the Commission Regulation EC no. 281/96 the proposed method has a decision limit of 109.1 µg l^− 1 and the capability of detection is 117.9 µg l^− 1 for both probability of false non-compliance and of false compliance equal to 5%. A recovery of 86.5% ± 2.4% (n = 5) has been obtained.