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.

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.

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.

Symmetric alteration of four knots of B-spline and NURBS surfaces

Modifying the knots of a B-spline curve, the shape of the curve will be changed. In this paper, we present the effect of the symmetric alteration of four knots of the B-spline and the NURBS surfaces, i.e., symmetrical alteration of the knots of surface, the extended paths of points of the surface will converge to a point which should be expressed with several control points. This theory can be used in the constrained shape modification of B-spline and NURBS surfaces.

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.

Ship hull plate processing surface fairing with constraints based on B-spline

The problem of ship hull plate processing surface fairing with constraints based on B-spline is solved in this paper. The algorithm for B-spline curve fairing with constraints is one of the most common methods in plane curve fairing. The algorithm can be applied to global and local curve fairing. It can constrain the perturbation range of the control points and the shape variation of the curve, and get a better fairing result in plane curves. In this paper, a new fairing algorithm with constraints for curves and surfaces in space is presented. Then this method is applied to the experiments of ship hull plate processing surface. Finally numerical results are obtained to show the efficiency of this method.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

Reconstruction of B-spline surface by interpolating boundary curves and approximating inner points

In this paper, we present an algorithm for reconstruction of B-spline surface such that it interpolates the four given bound- ary curves and simultaneously approximates some given inner points. The main idea of our method is： first, we construct an initial surface which interpolates the four given boundary curves; then, while keeping the boundary control points of the initial surface un- changed, we reposition the inner control points of the surface with energy optimization method. Examples show that our algorithm is practicable and effective.

G^0 and G^1 connection between two adjacent B-spline surfaces

In this paper, the smooth connection between two B-spline surfaces is discussed. First, a brief proof of some simple sufficient conditions of Go and G1 continuity is given. On this basis, a novel method for Go or G1 connection between two adjacent B-spline surfaces is presented. A reparameterization step is firstly taken for one of the surfaces such that they have the same parameterization in v direction, then, adjust their boundary control vertices to make them Go or Gl connected. The GI connection parameter is determined by an optimization problem. Compared with the existed methods, our method is simple and easy to be used in practice.

An Approach to Representing Heterogeneous Non-Uniform Rational B-Spline Objects

The representation method of heterogeneous material information is one of the key technologies of heterogeneous object modeling, but almost all the existing methods cannot represent non-uniform rational B-spline （NURBS） entity. According to the characteristics of NURBS, a novel data structure, named NURBS material data structure, is proposed, in which the geometrical coordinates, weights and material coordinates of NURBS heterogene- ous objects can be represented simultaneously. Based on this data structure, both direct representation method and inverse construction method of heterogeneous NURBS objects are introduced. In the direct representation method, three forms of NURBS heterogeneous objects are introduced by giving the geometry and material information of con- trol points, among which the homogeneous coordinates form is employed for its brevity and easy programming. In the inverse construction method, continuous heterogeneous curves and surfaces can he obtained by interpolating discrete points and curves with specified material information. Some examples are given to show the effectiveness of the pro- posed methods.

B-Spline Approximation of Ship Waves on the Free Surface

We consider the problem of a ship advancing in waves. In this method, the zone of free surface in the vicinity of body is discretized. On the discretized surface, the first-order and second-order derivatives of ship waves are represented by the B-Spline formulae. Different ship waves are approximated by cubic B-spline and the first and second order derivates of incident waves are calculated and compared with analytical value. It approves that this numerical method has sufficient accuracy and can be also applied to approximate the velocity potential on the free surface.

Extraction of Feature Points for Non-Uniform Rational B-Splines(NURBS)-Based Modeling of Human Legs

Methods of digital human modeling have been developed and utilized to reflect human shape features.However,most of published works focused on dynamic visualization or fashion design,instead of high-accuracy modeling,which was strongly demanded by medical or rehabilitation scenarios.Prior to a high-accuracy modeling of human legs based on non-uniform rational B-splines(NURBS),the method of extracting the required quasi-grid network of feature points for human legs is presented in this work.Given the 3 D scanned human body,the leg is firstly segmented and put in standardized position.Then re-sampling of the leg is conducted via a set of equidistant cross sections.Through analysis of leg circumferences and circumferential curvature,the characteristic sections of the leg as well as the characteristic points on the sections are then identified according to the human anatomy and shape features.The obtained collection can be arranged to form a grid of data points for knots calculation and high-accuracy shape reconstruction in future work.

G^1 Continuity Conditions of B-spline Surfaces

According to the B-spline theory and Boehm algorithm, this paper presents several necessary and sufficient G1 continuity conditions between two adjacent B-spline surfaces. In order to meet the need of application, a kind of sufficient conditions of G1 continuity are developed, and a kind of sufficient conditions of G1 continuity among N(N>2) patch B-spline surfaces meeting at a common corner are given at the end.

RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES

A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.

Fast B-spline surface extraction algorithm of seismic volume data on GPU

In the process of seismic data interpretation, the extraction of a horizon or a fault is generally needed. In this paper we present a fast extraction method. First select some feature points interactively, then reconstruct the surface according to the selected feature points by using B-spline interpolation curve or surface. In order to solve the error-drawing problem appeared in the process of interactive rendering, which is caused by the change of sampling interval as the view point changes, we combine shear-warp and splatting algorithm to render the surface. The rendering of seismic data and specific surface in our work are achieved on GPU platform using CUDA programming language, which make it able to meet the requirements of real-time rendering.

Modification of B-Spline Surfaces with Optimization

Algorithms of modifying a surface to approximate some scattered points, or pass through some characteristic points/curves are presented. Similar to variational approach, the algorithms are based on optimization. For the deviation between the modified surface and the original one is adopted as the objective functions, the change of the surface shape is as small as possible with the modified surface satisfying the specified requirements.