Research On Nonlocal Spline Theory

As a kind of effective approximation tool, the classical spline function is widely used in the field of computational geometry, differential equation numerical solution and computer aided design. At the same time, the spline function is closely related to the differential equation and mechanics. As a piecewise smooth polynomial function, its smoothness is defined by the classical calculus. With the study on s ome of the new materials more and more in-depth, the problem of breakage, dislocation,crack and the heat transfer and material internal micro scale effect will have an impact on the macro mechanical properties, consideration should be given to the nonlocal nature. Therefore, nonlocal field theory and calculus are created. In order to provide a new tool to solve the problems above, the nonlocal spline function theory is studied from the real physical background.Nonlocal spline function is for the promotion of classical spline function, starting from nonlocal field theory the nonlocal operator is defined, and then research the theory of the nonlocal spline function. First of all, we do research on the relationship between the nonlocal operator and divergence of differential operators in the nonlocal field theory, using divergence differential operator to express nonlocal operators, and de fine the generalized functions in the field of nonlocal operators, and simplify it to one-dimensional function field, two polynomial function is defined in the onedimensional function field; Then, with the same measure of definition of classical smoothness, we define nonlocal smoothness with nonlocal continuity either. We define nonlocal continuity on basis of which and combine with the physical background of nonlocal spline, parallel extensions of the classical spline function, gives the definition of a nonlocal spline function and its general expression,it goes back to classical spline function when the way of division of domain changes.The research content is a new attempt in the field of mathematics and mechanics. It is a new attempt to the theory of spline function. The results of the study not only make people have a deeper understanding of t he concept and expression of nonlocal sample, but also provide a new tool for solving the problem.