دانلود رایگان مقاله تقریب اسپلاین مکعب حفظ شکل صاف دو متغیره

Abstract

Given a piece-wise linear function defined on a type I uniform triangulation we construct a new partition and define a smooth cubic spline that approximates the linear surface and preserves its shape. The key piece is a new macro-element that has the ability to combine six independent gradients coming together at an interior vertex in a smooth yet shape-preserving fashion. The shape of the resulting spline surface follows local changes in the shape of the piece-wise linear interpolant without overshooting. We prove that convexity, positivity and monotonicity of the spline depend on the local data only. Computational scheme for Bernstein–Bezier spline coefficients is local and fast. Numerical examples highlight unique shape-preserving properties of the spline.

6. Conclusions

We have shown how to construct a triangulation ˜ that allows us to define a smooth cubic spline interpolating values and gradients of a piece-wise linear function at certain locations. The triangulation and the values to be interpolated are chosen to produce a solution that follows local changes in the behavior of the piece-wise linear function. The spline preserves monotonicity, convexity and positivity of the piece-wise linear function without overshooting. In fact, around each vertex of the initial triangulation, along certain directions the spline is variation diminishing. The scheme uses local data information, and simple explicit formulas for the coefficients of the spline are included. Numerical examples illustrate shape-preserving behavior of the spline and highlight visual differences induced by the choice of the parameter λ. Last example investigates the influence of λ on angles in quadrangles of ˜ and the resulting loss of accuracy.