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SUMMARYA ÿnite point method, least-squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisÿed not only at the collocation points but also at the auxiliary points in a least-squares sense. The moving least-squares interpolant is used to construct the trial functions. The computational e ort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution signiÿcantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational e ort.

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We have monitored the BL Lac object Mrk 501 in the optical V, R, and I bands from 2010 to 2015. For Mrk 501, the presence of a strong host galaxy component can affect the results of photometry. After subtracting the host galaxy contributions, the source shows intraday and long-term variabilities for optical flux and color indices. The average variability amplitudes of the V, R, and I bands are 22.05%, 22.25%, and 23.82%, respectively, and the value of the duty cycle is 14.87%. A minimal variability timescale of 106 minutes is detected. No significant time lag between the V and I bands is found on one night. The bluer-when-brighter (BWB) trend is dominant for Mrk 501 on intermediate, short, and intraday timescales, which supports the shock-in-jet model. For the long timescale, Mrk 501, in different states, can have different BWB trends. The corresponding results for non-correcting host galaxy contributions are also presented.

SUMMARYThe inherent no-slip contact constraint in the standard material point method (MPM) creates a greater penetration resistance. Therefore, the standard MPM was not able to treat the problems involving impact and penetration very well. To overcome these deficiencies, two contact methods for MPM are presented and implemented in our 3D explicit MPM code, MPM3D. In MPM, the impenetrability condition may not satisfied on the redefined regular grid at the beginning of each time step, even if it has been imposed on the deformed grid at the end of last time step. The impenetrability condition between bodies is only imposed on the deformed grid in the first contact method, while it is imposed both on the deformed grid and redefined regular grid in the second contact method. Furthermore, three methods are proposed for impact and penetration simulation to determine the surface normal vectors that satisfy the collinearity conditions at the contact surface. The contact algorithms are verified by modeling the collision of two elastic rings and sphere rolling problems, and then applied to the simulation of penetration of steel ball and perforation of thick plate with a particle failure model. In the simulation of elastic ring collision, the first contact algorithm introduces significant disturbance into the total energy, but the second contact algorithm can obtain the stable solution by using much larger time step. It seems that both contact algorithms give good results for other problems, such as the sphere rolling and the projectile penetration.

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The material point method (MPM) enhanced with B-spline basis functions, referred to as B-spline MPM (BSMPM), is developed and demonstrated using representative quasi-static and dynamic example problems. Smooth B-spline basis functions could significantly reduce the cell-crossing error as known for the original MPM. A Gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the BSMPM analysis of largedeformation problems for the improved accuracy and convergence, especially with the quadratic B-splines. Moreover, the increase in the order of the B-spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence. For plate impact examples, it is demonstrated that the BSMPM outperforms the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) methods in term of the accuracy of representing stress waves. Thus, the BSMPM could become a promising alternative to the MPM, GIMP, and CPDI in solving certain types of transient problems. KEYWORDS B-spline basis functions, Gauss quadrature, large deformation, material point method, mesh refinement, transient problems