Experimental study of core MHD behavior and a novel algorithm for rational surface detection based on profile reflectometry in EAST

Microwave reflectometry is a powerful diagnostic that can measure the density profile and localized turbulence with high spatial and temporal resolution and will be used in ITER,so understanding the influence of plasma perturbations on the reflect signal is important.The characteristics of the reflect signal from profile reflectometry,the time-of-flight(TOF)signal associated with the MHD instabilities,are investigated in EAST.Using a 1D full-wave simulation code by the Finite-DifferenceTime-Domain(FDTD)method,it is well validated that the local density flattening could induce the discontinuity of the simulated TOF signal and an obvious change of reflect amplitude.Experimental TOF signals under different types of MHD instabilities(sawtooth,sawtooth precursors and tearing mode)are studied in detail and show agreement with the simulation.Two new improved algorithms for detecting and localizing the radial positions of the low-order rational surface,the cross-correlation and gradient threshold(CGT)method and the 2D convolutional neural network approach(CNN)are presented for the first time.It is concluded that TOF signal analysis from profile reflectometry can provide a straightforward and localized measurement of the plasma perturbation from the edge to the core simultaneously and may be a complement or correction to the q-profile control,which will be beneficial for the advanced tokamak operation.

Judging or setting weight steady-state of rational Bézier curves and surfaces

Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.

Computation of lower derivatives of rational triangular Bézier surfaces and their bounds estimation

By introducing the homogenous coordinates, degree elevation formulas and combinatorial identities, also by using multiplication of Bernstein polynomials and identity transformation on equations, this paper presents some explicit formulas of the first and second derivatives of rational triangular Bézier surface with respect to each variable (including the mixed derivative) and derives some estimations of bound both on the direction and magnitude of the corresponding derivatives. All the results above have value not only in surface theory but also in practice.

Interval Bézier Surfaces Approximation of Rational Surfaces

On the basis of the perturbation, we present an approach to approximating rational surfaces by the interval Btzier surfaces using energy minimization method. The approach makes the perturbation surfaces have more restrictions than the original surfaces. It could be combined with subdivision method to obtain a piecewise interval polynomial approximation for a rational surface. The applications of this approach are illustrated too.

Visible region extraction from a sequence of rational Bézier surfaces

A method for computing the visible regions of free-form surfaces is proposed in this paper. Our work is focused on accurately calculating the visible regions of the sequenced rational Bézier surfaces forming a solid model and having coincident edges but no inner-intersection among them. The proposed method calculates the silhouettes of the surfaces without tessellating them into triangle meshes commonly used in previous methods so that arbitrary precision can be obtained. The computed sil- houettes of visible surfaces are projected onto a plane orthogonal to the parallel light. Then their spatial relationship is applied to calculate the boundaries of mutual-occlusion regions. As the connectivity of the surfaces on the solid model is taken into account, a surface clustering technique is also employed and the mutual-occlusion calculation is accelerated. Experimental results showed that our method is efficient and robust, and can also handle complex shapes with arbitrary precision.

STUDY ON THE METHOD OF CONSTRUCTING RATIONAL CUBIC CURVES AND CURVED SURFACE INCLUDING CONTROLLING PARAMETERS

By adopting the method of controlling parameters this paper describes the construction of various kinds of cubic curve segment and curved surface fragment with rational and non rational parameters, and discusses the relationship between controlling parameters, weighted factors and types, kinds and characteristics of curve segments and curved surface fragments. A mathematical method is provided for CAGD with abundant connotations, broad covering region, convenience, flexibility and direct simplicity.

Degenerations of rational Bézier surface with weights in the form of exponential function

Rational Bezier surface is a widely used surface fitting tool in CAD. When all the weights of a rational B@zier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bezier surfaces. In this paper, we study on the degenerations of the rational Bezier surface with weights in the exponential function and indicate the difference of our result and the work of Garcia-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bezier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.

Multivariate Rational Response Surface Approximation of Nodal Displacements of Truss Structures

Polynomial-basis response surface method has some shortcomings for truss structures in structural optimization,concluding the low fitting accuracy and the great computational effort. Based on the theory of approximation, a response surface method based on Multivariate Rational Function basis(MRRSM) is proposed. In order to further reduce the computational workload of MRRSM, focusing on the law between the cross-sectional area and the nodal displacements of truss structure, a conjecture that the determinant of the stiffness matrix and the corresponding elements of adjoint matrix involved in displacement determination are polynomials with the same order as their respective matrices, each term of which is the product of cross-sectional areas, is proposed. The conjecture is proved theoretically for statically determinate truss structure, and is shown corrected by a large number of statically indeterminate truss structures. The theoretical analysis and a large number of numerical examples show that MRRSM has a high fitting accuracy and less computational effort. Efficiency of the structural optimization of truss structures would be enhanced.

APPROXIMATION OF INTERVAL BEZIER SURFACES

Based on the conception of perturbation, an approach to the interval Bezier surfaces approximating ra- tional surfaces is presented using the energy minimization method. The method places more restrictions on the perturbation surfaces than the original surfaces. The applications of the approach are also presented. Experimen- tal result is combined with the subdivision method to obtain a piecewise interval polynomial approximation for a rational surface.

Incompressible Pairwise Incompressible Surfaces in Knot Exteriors

We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and let F be an incompressible pairwise incompressible surface in S 3-K. Then F is a punctured sphere.

Constrained Control of Interpolating Surfaces by Parameters

In order to meet the needs of practical design, an interpolation technique is employed to constrain the shape of surfaces. The method of preserving positivity on the interpolation surface and constraint on interpolating data is also developed. The advantage of this new method is that it can be used to constrain the shape of an interpolating surface only by selecting suitable parameters, and numerical examples are presented to show the performance of the method.

有理三角B-B曲面多项式逼近的一个有效算法

将美国计算机图形专家Sederberg提出的有理曲线多项式逼近的思想与算法推广到工程中广泛采用的三角域上的有理曲面.主要工作是:给定一张有理三角B-B曲面,通过将多项式三角B-B曲面的控制顶点表示成相同次数的有理三角B-B曲面的形式,即将多项式曲面的移动控制顶点看作在有理三角B-B曲面上的移动点,并添加约束条件,构造了三角域上的Hybrid曲面;适当地选取有理三角B-B曲面的1次Hybrid曲面表示,推导了彼此等同但次数相邻的两张Hybrid曲面之间控制顶点的递推公式;利用Hybrid曲面移动控制顶点凸包内的一点来代替该移动控制顶点,得到了多项式三角B-B曲面逼近有理三角B-B曲面的一个算法,并在文中给出了数值实例.这些结果可以明显地提高计算机辅助几何设计系统的数据可换性与计算效率.

Determination and(Re)Parametrization of Rational Developable Surfaces

The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors first determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affrmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface.

Lattices,Diophantine equations and applications to rational surfaces

In this article,we study certain quadratic Diophantine equations in Picard lattices of blow-ups of the complex projective plane,and describe their relations with root systems and Weyl group orbits of quasiminuscule fundamental weights.We apply these to study the geometry of certain rational surfaces.

The minimal genus problem in rational surfaces CP^2#nCP^2

There are three key ingredients in the study of the minimal genus problem for rational surfaces CP2#nCP2: the generalized adjunction formula, the action of the orthogonal group of the Lorentz space and the geometric construction. In this paper, we prove the uniqueness of the standard form (see Definition 1.1 and Theorem 1.1) of a 2-dimensional homology class under the action of the subgroup of the Lorentz orthogonal group that is realized by the diffeomorphisms of CP2#nCP2.Using the geometric construction, we determine the minimal genera of some classes (see Theorem 1.2).

A Subdivision Scheme for Rational Triangular Bézier Surfaces

An explicit formula is developed to decompose a rational triangular Bezierpatch into three non-degenerate rational rectangular B6zier patches of the samedegree. This formula yields a stable algorithm to compute the control verticesof those three rectallgular subpatches. Some properties of the subdivision arediscussed and the formula is illustrated with an example.

EXACT SOLUTIONS FOR A SURFACE WAVE EQUATION

Some exact travelling wave solutions and rational travelling wave solutions of a surface wave equation in a convecting fluid are given in this paper.

A Survey of the Representations of Rational Ruled Surfaces

The rational ruled surface is a typical modeling surface in computer aided geometric design.A rational ruled surface may have different representations with respective advantages and disadvantages.In this paper,the authors revisit the representations of ruled surfaces including the parametric form,algebraic form,homogenous form and Plucker form.Moreover,the transformations between these representations are proposed such as parametrization for an algebraic form,implicitization for a parametric form,proper reparametrization of an improper one and standardized reparametrization for a general parametrization.Based on these transformation algorithms,one can give a complete interchange graph for the different representations of a rational ruled surface.For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces,the characterization methods are recalled to identify the ruled surfaces from them.

CHARACTERISATION OF RATIONAL AND NURBS DEVELOPABLE SURFACES IN COMPUTER AIDED DESIGN

In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions∧,M,ν.Properties of developable surfaces are revised in this framework.In particular,a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions∧,M,ν,which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative.It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant∧,M,ν.The results are readily extended to rational spline developable surfaces.

The Geometric Continuity of Rational Bezler Triangular Surfaces

The problems of geometric continuity for rational Bezter surfaces are discussed. Concise conditions of first order and second order geometric continuity for rational triangular Bézier surfaces are given. Meanwhile,a geometric condition for smoothness between adjacent rational Bézier surfaces and the transformation formulae between rational triangular patches and rational rectangular patches are obtained.