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As a well-known numerical approximation method, the least squares fitting method is widely applied in surveying engineering. The most important and difficult problem in this method is to determine the type of base function to be used, and polynomial is the most frequently used type. But if the degree of polynomial is too high (>=7), the normal equations are usually ill-conditioned. When we choose the uniform B-spline as the basis, there are two problems: (1) if we require the known data points to be an equidistant grid, it can only be applied in limited field; and (2) if the known data points are scattered, it is very difficult to obtain satisfactory results by using the uniform B-spline method. The purpose of this paper is to present the least squares fitting method based on a bivariate nonuniform B-spline. The principles and algorithms of this least squares fitting method are developed. Based on this method, two numerical simulations and one application in regional gravity field approximation are discussed. [ABSTRACT FROM AUTHOR]

The effect of samarium (Sm) addition on the microstructure, mechanical performance and corrosion behavior of extruded AZ41- x Sm (x = 0%, 1%, 2% and 3%) alloys has been investigated by using optical microscope (OM), X-ray diffusion (XRD), scanning electronic microscope (SEM) equipped with energy dispersive spectroscope (EDS), tensile testing, hydrogen evolved and electrochemical tests. It is observed that Sm alloyed AZ41 alloys consist of α -Mg substrate, β -Mg 17 Al 12 and Al 2 Sm intermetallic. The β -phase is gradually inhibited with increasing Sm content. The tensile testing result shows that the appropriate amount of Sm addition slightly enhances the mechanical strength. 2Sm alloyed alloy (AZ41-2Sm) displays optimal tensile properties, with a yield tensile strength (YTS) of 216.4 MPa, ultimate tensile strength (UTS) of 295.9 MPa and elongation (EL) of 15.9%, which is the result of second phase strengthening. The addition of Sm significantly improved the corrosion resistance of AZ41 alloy. The AZ41-1Sm alloy exhibits the best corrosion resistance properties with a corrosion rate about one-half that of the AZ41 alloy based on the hydrogen evolved and electrochemical test. • Al 2 Sm is formed in the as-extruded AZ41 alloy with Sm addition. • The AZ41 alloy is coarsened by the addition of 1–2% Sm, but refined by 3% Sm. • The mechanical properties of AZ41 alloy are enhanced slightly by the addition of Sm. • The AZ41-1Sm alloy shows the optimal corrosion resistance of the AZ41-xSm alloys. [ABSTRACT FROM AUTHOR]