Research On Bending Energy Of Mesh Multivariate Spline Surfaces And Its Application

Spline function can be understood as a certain smoothness segment(slice) polynomial, and it is the fundamental tool for curve and surface modeling. Multivariate spline is the extension of univariate spline, and it can be roughly divided into two directions: one is that reserve the properties of piecewise spline, and research the smooth connection between adjacent surface and overall coordination, thus establishing the basic theory of multi-grid splines; the second is to bend the surface energy functional as the objective function for constrained optimization, satisfy the interpolation conditions, and get thin plate spline function, which is the physical energy point of view on the multiple promotion. The mechanical meaning and the bending energy of multiple grid surface remains to be further research. From the mechanical meaning of spline, the bending energy of multivariate spline surfaces and its application in surface modeling are researched.Firstly, the beam is deformed through exerting concentrated couple on cantilever beam and the overhanging beam, the formed flexural meets the piecewise smooth and constitutes the spline function of univariate quadric spline, and the mechanical structure principle of quadric spline function in boundary constraints is achieved. The bending energy functional of univariate spline is researched, using the variational principle and the principle of minimum potential energy boundary conditions of beams under different circumstances and deflection equation is obtained, and then the connection between the math expression of the spline function and the engineering mechanics is established. Secondly, by exerting couple and the bending moment in equilibrium distribution on the plate boundary and subdivision line, we make the pure bending deformation of plate on the deflection surface subdivision of rectangle subdivision have shard form, becoming the grid spline surface of bivarite quadric spline, the mechanical model and energy functional of multivariate spline on a given rectangular grid are studied. Then for the arbitrary shape of fixed edge plate, generalized energy and the boundary conditions are obtained. Finally, in combination with optimization theory, energy optimization method for surface modeling is proposed, and surface modeling method under the constraint condition of the boundary curves and parameters surface are given.This study belongs to the cross field of mathematics and mechanics, the research results not only make people have a deeper understanding of the mechanical meaning and bending energy of the multivariate spline, also are to supplement and complete the theory of spline function, and provide new ideas for energy constraint method of the complex curve surface modeling. Figure16; Table0; Reference 68...