GALERKIN MESHLESS METHODS BASED ON PARTITION OF UNITY QUADRATURE

Numerical quadrature is an important ingredient of Galerkin meshless methods. A new numerical quadrature technique, partition of unity quadrature （PUQ）,for Galerkin meshless methods was presented. The technique is based on finite covering and partition of unity. There is no need to decompose the physical domain into small cell. It possesses remarkable integration accuracy. Using Element-free Galerkin methods as example, Galerkin meshless methods based on PUQ were studied in detail. Meshing is always not required in the procedure of constitution of approximate function or numerical quadrature, so Galerkin meshless methods based on PUQ are “truly” meshless methods.

Incompatible numerical manifold method for fracture problems

The incompatible numerical manifold method （INMM） is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numerical manifold method employs two cover systems as follows. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the weighted functions, and the physical cover system describes geometry of the domain and the discontinuous surfaces therein. In INMM, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the process of displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the INMM is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To improve the precision of the INMM, the analytical solution is used at the tip of a crack, and thus the cover displacement functions are extended with higher precision and computational efficiency. Some numerical examples are given.

THEORETICAL STUDY OF THREE-DIMENSIONAL NUMERICAL MANIFOLD METHOD

The three-dimensional numerical manifold method（NMM） is studied on the basis of two-dimensional numerical manifold method. The three-dimensional cover displacement function is studied. The mechanical analysis and Hammer integral method of three-dimensional numerical manifold method are put forward. The stiffness matrix of three-dimensional manifold element is derived and the dissection rules are given. The theoretical system and the numerical realizing method of three-dimensional numerical manifold method are systematically studied. As an example, the cantilever with load on the end is calculated, and the results show that the precision and efficiency are agreeable.

Virtual homological eigenvalues and the Weil-Petersson translation length

For any pseudo-Anosov automorphism on an orientable closed surface,an inequality is established by bounding certain growth of virtual homological eigenvalues with the Weil-Petersson translation length.The new inequality fits nicely with other known inequalities due to Kojima and McShane(2018)and Lê(2014).The new quantity to be considered is the square sum of the logarithmic radii of the homological eigenvalues(with multiplicity)outside the complex unit circle,called the homological Jensen square sum.The main theorem is as follows.For any cofinal sequence of regular finite covers of a given surface,together with lifts of a given pseudo-Anosov,the homological Jensen square sum of the lifts grows at most linearly fast compared with the covering degree,and the square root of the growth rate is at most 1/√4πtimes the Weil-Petersson translation length of the given pseudo-Anosov.