Some Researches On The Mechanical Meaning Of Multivariate Splines And Adaptive Data Analysis

In 1946,I.J.Schoenberg established the basic theory of unvariate spline systematically, and pointed out the mechauical viewpoint of spline functions,that is,a cubic spline s(x) corresponds to the deflection curve of an ordinary(infinite) beam under the action of suitable concentrated loads,where the name "spline function" is from.Based on the above viewpoint and principle of minimum potential energy,J.C.Holladay showed that a natural cubic spline is the smoothest curve which minimizes the potential energy in 1957. So there are natural relations between splines and mechanics.In 1977,J.Duchon generalized splines from 1D to nD based on constrained optimization,and got the so called "thin-plate spline",a radial basis function without piecewise features,which minimizes the bending energy of a thin plate and satisfies the interpolation conditions.In 1975,R. H.Wang intrduced the "smoothing cofactor" and "conformality conditions" by analyzing the smoothness and divisibility of two polynomials over the two adjacent cells,and created the so called "algebraic geometry method" to study the basic theory of multivariate splines on arbitrary partitions.Fruitful achievements of multivariate splines have been obtained after more than 30 years' rapid developments,but the relations between multivariate splines and mechanics have not been established.With the widely applications of splines in the fields such as FEM,CAGD,function approximation,computer graphics, data analysis and so on,to establish the mechanical background of multivariate splines becomes a meaningful and worth studying work.In Chapter 2,we established the relationships between multivariate splines and mechanics based on the theory of elasticity and theory of plates and shells.The loads acting on the plates involved couples,distributed loads and concentrated loads;the partitions involved rectangular partition,type-1 triangulations and general three direction meshes. Which just respond the complexity of multivariate problems.Main results are as follows:Making use of mechanical analysis method,by acting couples along the interior edges with suitable evaluations,the deflection surface was divided into piecewise form,therefore, the relation between a class of bivariate splines on rectangular partition and the pure bending of thin plate is established.In addition,the interpretation of Smoothing Cofactor and Conformality Condition from the mechanical point of view is given.Furthermore, by introducing twisting moments,the mechanical background of any spline belong to the above space is set up.We established the relationship between a class of bivariate splines on three direction meshes and the bending of simply supported polygonal thin plate.In this case,we break the restriction of external loads must be constant quantity and take them as linear function of x,y,i.e.,M_n=M_n(x,y).This result is equivalent to the deflection surface of a uniformly stretched membrane acted by suitable distributed loads.From the relation between univariate quartic C~3 spline on equidistant knots and beam bending acted by piecewise uniformly distributed loads,generalizing to 2D,we got a special kind of bivariate splines over regular triangulation which implies the golden section in the piecewise algebric curves determined by them.We also discussed the variational property of bivariate splines.Hilbert-Huang Transform(HHT) is an adaptive data analysis method for nonlinear and non-stationary data,which consists of two parts:Empirical Mode Decomposition(EMD) and Hilbert spectral analysis(HSA).The basis of HHT is not prior determined as Fourier analysis,but generated by EMD based on and derived from the data,which is called Intrinsic Mode Functions(IMF).The HHT's power and effectiveness in data analysis have been demonstrated by its successful application to many important problems. But it is entirely empirical and suffers from a lack of generally accepted theoretical framework. To set up the mathematical foundation of HHT is an open problem need to be solved emergently.Chaper 3 investigates random data analysis by EMD.First,basing on the bending and stochastic vibration of beam,further research and the mecbanical meaning of random splines were given.Owing to EMD not sufficiently consider the randomness of data from nonlinear and non-stationary systems,we try to use random spline to construct the mean envelope and analyze data adaptively in terms of statistics.The random-EMD method was presented and some numerical simulations were demonstrated.Chapter 4 analyzes the bidimensional IMFs based on the relations between bivariate splines and bending of thin plates.The definitions of bidimensional IMF and weak-IMFs were given.The relations between bidimensional IMFs and vibration of thin plates or a class of PDE were tried to be established.