Study On Partition Of Unity Method In Solid Mechanics

During the last ten years, the Meshless or Meshfree Methods have been a focus of research in the field of computational mechanics. Among several kinds of Meshless Methods, Partition of Unity Method(PUM) is very flexible in constructing local trial (and test) spaces and constructing a partition of unity. This thesis does some research about problems of crack and large deformation of beam, about how flexibility of PUM influences accuracy and convergence of numerical results, and draws some valuable conclusions.This thesis begins with the construction of local trial spaces. Common numerical methods, such as FEM, BEM and most Meshless Methods, construct local trial spaces with polynomials. Facing problems of high singularity, polynomials performs poorly in converging to exact solution; PUM perfoms well by incorporating enriched functions into local trial spaces. The thesis gives examples of Possion Problem whith high gradient and plane crack problem, in which local trial spaces have incorporated enriched functions about those examples. Enriched functions usually give a prior kon- wledge about the problem under consideration. Numerical results show that PUM can improve the accuracy and convergence well with enriched functions.The thesis does some research about problems of large deformation of beam. Partition of unity includes Shepard function and shape function of FEM. Shepard func -tion is constructed with Gauss Weighted function and Cubic Spline funcition. Numeri -cal results show that PUM can performs well even when mesh of FEM is excessively deformed or nodes of Meshless Methods are irregular, can converge to exact solution quickly. Results are influenced by parameters when Shepard functions are constructed by Gauss Weighted function, but when Cubic Spline function is used, numerical results give more stability.