Implementation of a Conservative Two-step Shape-Preserving Advection Scheme on a Spherical Icosahedral Hexagonal Geodesic Grid

An Eulerian flux-form advection scheme, called the Two-step Shape-Preserving Advection Scheme （TSPAS）, was generalized and implemented on a spherical icosahedral hexagonal grid （also referred to as a geodesic grid） to solve the transport equation. The C grid discretization was used for the spatial discretization. To implement TSPAS on an unstructured grid, the original finite-difference scheme was further generalized. The two-step integration utilizes a combination of two separate schemes （a low-order monotone scheme and a high-order scheme that typically cannot ensure monotonicity） to calculate the fluxes at the cell walls （one scheme corresponds to one cell wall）. The choice between these two schemes for each edge depends on a pre-updated scalar value using slightly increased fluxes. After the determination of an appropriate scheme, the final integration at a target cell is achieved by summing the fluxes that are computed by the different schemes. The conservative and shape-preserving properties of the generalized scheme are demonstrated. Numerical experiments are conducted at several horizontal resolutions. TSPAS is compared with the Flux Corrected Transport （FCT） approach to demonstrate the differences between the two methods, and several transport tests are performed to examine the accuracy, efficiency and robustness of the two schemes.

ON SHAPE PRESERVING EXTENSION AND APPROXIMATION OF FUNCTIONS

We establish the concept of shapes of functions by using partial differential inequalites. Our definition about shapes includes some usual shapes such as convex,subharmonic,etc.,and gives many new shapes of functions.The main results show that the shape preserving approxi- mation has close relation to the shape preserving extension.One of our main results shows that if f∈C(Ω)has some shape defined by our definition,then f can be uniformly approximated by polynomials P_n ∈p_n(n∈N)which have the same shape in Ω,and the degree of the ap- proximation is Cω(f,n^(-β))with constants C,β>0.

SHAPE PRESERVING QUADRATIC SPLINEINTERPOLATION

A new method for constructing quadratic spline to interpolate a given sat of data points ispresented. The constructed spline preserves the shape of the given data points such as monotonicityand convexity , and is visually pleasing. Numerical experiments are included which compare the ″visu-ally pleasing″ and the approximation accuracy of the new method with other two methods.

SHAPE PRESERVING INTERPOLATION USING PIECEWISE CUBIC

The papcr gives an approach to construct shape preserving piece wise cubic, where twocubic picces are allowed by inscrting at most a new knot in each data subinterval, and these cubicpieees is C2 continuous at each new knot. Two numerical examples show that the method is effec.tive and visually pleasing.

ESTIMATES OF LINEAR RELATIVE n-WIDTHS IN L^p [0,1]

In this paper we will show that if an approximation process {Ln}n∈N is shape- preserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1], and the operators Ln are assumed to be of finite rank n, then the order of convergence of D^kLnf to D^kf cannot be better than n-2 even for the functions x^k, x^k＋1, x^k＋2 on any subset of [0,1 ] with positive measure. Taking into account this fact, we will be able to find some asymptotic estimates of linear relative n-width of sets of differentiable functions in the space LP[0, 1], p ∈ N.

Total Positivity of UE Spline Basis

A new geometric method to prove the total positivity of UE spline basis was proposed. UE spline basis is a kind of basis defined over algebraic-trigonometric unified space. UE spline basis shares most properties of the usual polynomial B-Splines. Total positivity is an important property for spline basis, it is highly related with shape preserving and variation diminishing properties. Knot inserted algorithm is the most useful algorithm for spline curves since many other useful properties are based on it. It is necessary to prove the total positivity of UE spline basis using knot inserted algorithm intuitively, not only enrich the UE spline basis theory, but also can be treated as supplement to the total positivity in algebraic sense. This approach also can be extended to other analogical bases.

BEST LIPSCHITZ CONSTANTS FOR THE BEZIER NETS AND BERNSTEIN POLYNOMIALS OVER A SIMPLEX

The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves that (1) if f(x) is Lipschitz continuous over sigma, i.e., f(x) is an element of Lip(A)(alpha,sigma), then both the n-th Bezier net <(f)over cap (n)> and the n-th Bernstein polynomial B-n(f;x) corresponding to f(x) belong to Lip(B)(alpha,sigma) , where B = Asec(alpha)Phi; and (2) if n-th Bezier net <(f)over cap (n)> is an element of Lip(A)(alpha,sigma), then the elevation Bezier net <E(f)over cap (n)> and the corresponding Bernstein polynomial. B-n(f,;x) also belong to Lip(A)(alpha,sigma). Furthermore, the constant B = Asec(alpha)Phi, in case (1) is best in some sense.

AN APPROACH TO EMPIRICAL DATA PREPROCESSING FOR BOTH CURVES AND SURFACES AND ITS APPLICATIONS

Some local algorithms for the,preprocessing,of empirical data for both curves and surfaces are proposed so that the modified data,a polygon for curve data and a triangulation for surface data,can be used not only to approximate the given data with required accuracy,but also to preserve the shape and geometric and mathematical properties of the data such as convexity and monotonicity etc.These algorithms would produce small errors (O(h￣4)) if the initial data are accurate and reasonably distributed.

Enlarging Image by Constrained Least Square Approach with Shape Preserving

From a visual point of view, the shape of an image is mainly determined by the edges. Conventional polynomial interpolation of image enlarging methods would produce blurred edges, while edge-directed interpolation based methods would cause distortion in the non-edge areas. A new method for image enlarging is presented. The image is enlarged in two steps. In the first step, a fitting surface is constructed to interpolate the image data. To remove the zigzagging artifact for each pixel, a fitting patch is constructed using edge information as constraints. The combination of all the patches forms the fitting surface which has the shape suggested by image data. Each point on the fitting surface can be regarded as a sampling point taken from a unit square domain, which means that when the fitting surface is used to enlarge the image, each sampling domain of the enlarged pixels is also a unit square, causing the enlarged image to lose some details. To make the enlarged image keep the details as many as possible, the sampling domain of the enlarged pixels should be less than a unit square. Then, in the second step, using the points taken from the fitting surface, new pixels are computed using constrained optimization technique to form the enlarged image, and the size of the sampling domain of the enlarged pixels is inversely proportional to the size of the enlarged image. The image enlarged by the new method has a quadratic polynomial precision. Comparison results show that the new method produces resized image with better quality.

Shape preserving topology optimization for structural radar cross section control

The purpose of this paper is to present a shape preserving topology optimization method to prevent the adverse effects of the mechanical deformation on the Radar Cross Section(RCS).The optimization will suppress the variation of RCS on the perfect conductor surface by structural design.On the one hand,the physical optics method is utilized to calculate the structural RCS,which is based on the surface displacement field obtained from the finite element analysis of the structure.The corresponding design sensitivities of topology optimization are derived analytically and solved by the adjoint method.On the other hand,the RCS variation and mechanical performance are taken into account simultaneously by extending a standard compliance-based topology optimization model.Two optimization formulations are discussed in an illustrative example,where the influences of upper limits of the compliance and the RCS variation are considered.Two more examples are further tested to show the ability and validity of the proposed optimization method.

Determining Knots by Minimizing Energy

A new method for determining knots to construct polynomial curves is presented. At each data point, a quadric curve which passes three consecutive points is constructed. The knots for constructing the quadric curve are determined by minimizing the internal strain energy, which can be regarded as a function of the angle. The function of the angle is expanded as a Taylor series with two terms, then the two knot intervals between the three consecutive points are defined by linear expression. Between the two consecutive points, there are two knot intervals, and the combination of the two knot intervals is used to define the final knot interval. A comparison of the new method with several existing methods is included.