Research On Spline Theory Based On Mechanical Analysis

Spline function has been widely used in the design of surface modeling. The spline functionoriginated from the small deflection bending deformation of a thin wood, which is a mathematicsimulation of deformation curve. So, the spline function is rooted in the actual spline curve, whosebackground is engineering mechanics. Multivariate spline is the extension of univariate spline.Professor Wang Renhong is the first person who utilizes algebraic geometry method to study thesmooth connection between adjacent surfaces, and then present "smooth cofactor coordinate method".The piecewise algebraic surface theory of multivariate spline under various subdivision is built in thissituation. When energy functional is acted as the objective function for constrained optimization,together with the interpolation condition is satisfied, and then the spline function of thin plate can beobtained. And this is physical energy view of multivariate spline. Both univariate spline andmultivariate spline associate closely with mechanics. With the further study of spline frommathematical theory, the actual mechanical meaning of spline especially the relationship betweenmultivariate spline value and mechanics needs further research.Firstly, connect the theory of spline function and geometric modeling method with themechanical analysis through exerting concentrated couple on cantilever beam and the overhangingbeam, the result of which is the beam bended and deformed. The formed flexural joins the piecewisesmooth together and constitutes the spline function of univariate quadric spline. The mechanicalstructure principle of quadric spline function in boundary constraints is achieved. And, the connectionbetween the math expression of the spline function and the engineering mechanics is established. Byexerting couple and the bending moment in equilibrium distribution on the plate boundary andsubdivision line, make the pure bending deformation of plate on the deflection surface subdivision ofrectangle subdivision have shard form, becoming the grid spline surface of bivarite quadric spline.Through changing the form of torque and subdivision, the piecewise polynomial surface with differenttimes and smoothness under special boundary condition can be gained, focusing on the shard side,mechanical model and energy functional of multivariate spline on a given grid. The results of theresearch not only offer a deeper understanding about the mechanical meaning of the spline, but alsoimprove and enrich the theory of spline function. At last, a new thought and method of solving theproblem of complex curved surface and solid modeling is provided.