Numerical Simulations of Electromagnetic Scattering from 2D Rough Surface: Geometric Modeling by Nurbs Surface

Induced current distributions on conducting bodies of arbitrary shape modelled by NURBS (non uniform rational B-splines) surfaces are obtained by using a moment method approach to solve an electric field integral equation (EFIE). The NURBS surfaces are expanded in terms of Bezier patches by applying the Cox-de Boor transformation algorithm. This transformation is justified because Bezier patches are numerically more stable than NURBS surfaces. New basis functions have been developed which extend over pairs of Bezier patches. These basis functions can be considered as a generalization of "rooftop" functions. The method is applied to obtain RCS values of several objects modelled with NURBS surfaces. Good agreement with results from other methods is observed. The method is efficient and versatile because it uses geometrical modelling tools that are quite powerful.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper introduces the nonuniform rational B-spline (NURBS) surfaces to improve the geometric modeling of the higher order method of moments (MoM). The electric field integral equation (EFIE) is discretized by the hierarchical higher order basis functions and converted to a matrix equation. Then the elements of the impedance matrix are efficiently evaluated by a new set of formulas. The bistatic radar cross sections (RCS) obtained by this new technique are compared with those obtained by the commonly used higher order MoM. The example of a cylinder and a missile shows excellent accuracy of the NURBS surfaces and that of the resultant RCS. Moreover, this new technique can fully exploit the flexibility of the higher order basis functions when the surface is highly curved, whereas the commonly used higher order MoM can not.

This paper focuses on the design of the extended branch structure for rapidly and simultaneously interpolating non-uniform rational B-spline (NURBS) surfaces and their partial derivatives on CNC machines. NURBS surfaces are concerned in the applications of molding process. However, the copious and complicated operations usually limit the applications of NURBS surfaces in CNC machining systems. In this paper, by considering the derivatives of basis functions, an extended branch structure is derived such that the computation time can be significantly reduced in computing NURBS surfaces and their partial derivatives. According to the machining results on a 3-axis vertical machining center, the extended branch structure is feasible to implement on CNC machines and is more efficient than existing approaches in computation.

We develop a realistic computerized torso phantom for use in medical imaging research. The phantom has the potential to model anatomical variations and patient motion more realistically than phantoms based on simple geometric primitives. Structures are defined basing nonuniform rational B-spline (NURBS) surfaces, as used in 3D computer graphics. By fitting NURBS to patient data, the phantom is more realistic than those based on solid geometry. Points on the surfaces of the different structures in the torso except the heart were selected from the Visible Human Project CT data set, National Library of Medicine. Polygon surfaces were fit to the points for each structure and smoothed. 3D NURBS surfaces were fit to the smooth polygon surfaces. The heart model was generated with 4D NURBS as described in a previous study. The control points defining each NURBS surface can be manipulated to model anatomical variations and patient motion.

An accurate creeping ray-tracing algorithm is presented in this paper to determine the tracks of creeping waves (or creeping rays) on arbitrarily shaped free-form parametric surfaces [nonuniform rational B-splines (NURBS) surfaces]. The main challenge in calculating the surface diffracted fields on NURBS surfaces is due to the difficulty in determining the geodesic paths along which the creeping rays propagate. On one single parametric surface patch, the geodesic paths need to be computed by solving the geodesic equations numerically. Furthermore, realistic objects are generally modeled as the union of several connected NURBS patches. Due to the discontinuity of the parameter between the patches, it is more complicated to compute geodesic paths on several connected patches than on one single patch. Thus, a creeping ray-tracing algorithm is presented in this paper to compute the geodesic paths of creeping rays on the complex objects that are modeled as the combination of several NURBS surface patches. In the algorithm, the creeping ray tracing on each surface patch is performed by solving the geodesic equations with a Runge-Kutta method. When the creeping ray propagates from one patch to another, a transition method is developed to handle the transition of the creeping ray tracing across the border between the patches. This creeping ray-tracing algorithm can meet practical requirements because it can be applied to the objects with complex shapes. The algorithm can also extend the applicability of NURBS for electromagnetic and optical applications. The validity and usefulness of the algorithm can be verified from the numerical results.

DN-Edit is a virtual environment developed to allow the manipulation of non-uniform rational b-spline (NURBS) surfaces using virtual shaping tools. NURBS have become the industry standard for representation of free-form curves and surfaces. The contribution of the work presented here is in the development of shaping tools which are used to operate directly on the NURBS data and change the shape of the surfaces in a virtual environment. These shaping tools allow surface manipulations to be made using methods that match the shaping of real malleable objects. The virtual shaping tools are three-dimensional shapes that are controlled through a six degree-of-freedom tracking system that converts user hand motions into computer input. The NURBS surface updates itself in real time due to the effect of the tools on the surface. The new shape of the surface is dependent on the position and orientation of the shaping tool relative to the surface. Constraint-based surface manipulation is used to obtain multiple point direct manipulation of NURBS surfaces. In addition, computational methods to allow the user to have direct control of the first derivative of the surface over an area are implemented. This application has been developed using the C2 software libraries and Iowa State University’s C2 surround screen virtual environment.

Chapter 10 - Subdivision Surfaces

Physical optics (PO) requires the calculation of a surface integral on a structure, which can be called as the PO integral. The surface of most structures can be accurately represented by nonuniform rational B-spline (NURBS) surfaces. In this letter, the PO integral over NURBS surfaces is reduced to an exact line integral by using a Radon transform (RT) interpretation. The integration line is shown to be the time-dependent intersection line of the NURBS surface of interest with a plane determined by the illumination and observation directions. Hence, the accuracy of evaluating the PO integral is limited only by the geometrical evaluations involving the NURBS surfaces. The accuracy of the derived closed form expression is demonstrated through numerical examples.

The multivariate quartic NURBS surfaces

In this work we present the optical design of an instrument for measuring the scattering of light by a 2D rough surface in the visible between 550nm and 830nm. An elliptical mirror is used to collect the light scattered by a 2D-surface. The 2D-surface is placed at the first focus of the ellipse. A relay is placed at the second focus of the ellipse to collect the light scattered by the surface in each direction, that is, parallel rays leaving the illuminated 2D surface at an arbitrary angle are brought to a point at the focal plane of the relay. The light is detected by a CCD camera. To cover 180 degrees field of view it is necessary to move the CCD linearly in two directions in the focal plane.

This paper introduce the Nonuniform Rational B‐Spline (NURBS) surfaces to improve the geometric modeling of the Locally Corrected Nyström (LCN) Method.Compared with the interpolation curvilinear quadrilaterals, the NURBS surfaces can approximate the model with higher accuracy and by much less elements. Thus, the size of the elements can be more flexibly chosen to fully exert the efficiency of the LCN method. Numerical results of the radar cross sections (RCS) show the validity of the proposed technique. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26034

Cultural heritage’s structural changes and damages can influence the mechanical behaviour of artefacts and buildings. The use of finite element methods (FEM) for mechanical analysis is largely used in modelling stress behaviour. The workflow involves the use of CAD 3D models and the use of non-uniform rational B-spline (NURBS) surfaces. For cultural heritage objects, altered by the time elapsed since their creation, the representation created with the CAD model may introduce an extreme level of approximation, leading to wrong simulation results. The focus of this work is to present an alternative method intending to generate the most accurate 3D representation of a real artefact from highly accurate 3D reality-based models, simplifying the original models to make them suitable for finite element analysis (FEA) software. The approach proposed, and tested on three different case studies, was based on the intelligent use of retopology procedures to create a simplified model to be converted to a mathematical one made by NURBS surfaces, which is also suitable for being processed by volumetric meshes typically embedded in standard FEM packages. This allowed us to obtain FEA results that were closer to the actual mechanical behaviour of the analysed heritage asset.

A novel implementation of the characteristic basis function method (CBFM) is given, in which the high-level basis functions, called characteristic basis functions (CBFs), are represented in terms of curved rooftops generated from nonuniform rational B-splines (NURBS) in the parametric (u,v) domain. The associated macro-testing functions are defined by using curved razor-blade functions corresponding to each rooftop. The underlying objective of the CBFM is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments (MoM). The result is, therefore, an approach which can handle many complex cases via direct solvers, without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill-conditioned matrices. As a result of the combination of the CBFM with the special class of low-level basis and testing functions directly located over NURBS surfaces, complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the CPU time as well as required memory resources.

An improved hybrid method of moments and physical-optics (MOM–PO) method is presented to analyse an antenna around an electrically large platform modelled with non-uniform rational B-spline (NURBS) surfaces. A Ludwig integral combined with stationary phase method is used to calculate the integral of induced currents for overcoming the limitations of the existing NURBS MOM–PO method. Grouping of the MOM region is utilised to enhance computational efficiency. Comparison of the results obtained from this method with that from MOM and existing NURBS MOM–PO shows that the present method is feasible.

본 연구에서는 최적화기법과 전산유체역학의 기술을 이용하여 저항의 관점에서 최적의 형상을 가지는 선형을 개발하는 알고리즘을 개발하였다. 최적화기법으로는 SQP(sequential quadratic programming)을 사용하였으며, 목적함수인 저항을 구하기 위하여 먼저 조파저항은 비선형 자유수면 경계조건을 고려한 선체주위 포텐셜유동을 계산할 수 있는 수치해석기법인 상 방향 패널이동법을 사용하였고, 선체에 미치는 전 저항을 구하기 위하여 ITTC 1957년 모형선-실선상관곡선을 이용하였다. 선형최적화 과정 중의 선체의 변경이나 계산 격자의 생성은 NURBS(Non-Uniform Rational B-Spline)기법을 사용하여 구현하였다. 이와 같은 방법을 사용하여 개발된 선형최적화 기법의 타당성을 검증하기 위하여 선형이 비교적 잘 알려진 건형인 Wigley선형과 Series 60(CB=0.6)선형에 대하여 설계속도 Fn=0.316에서 선형최적화를 위한 수치해석을 수행하고 그 결과를 초기선형과 서로 비교하였다. This paper presents the method for developing an optimum hull form with minimum wave resistance using SQP(sequential quadratic programming) as an optimization technique. The wave resistance is evaluated by a Rankine source panel method with non-linear free surface conditions and the ITTC 1957 friction line is used to predict the frictional resistance coefficient. The geometry of the hull surface is represented and modified using NURBS(Non-Uniform Rational B-Spline) surface patches. To verity the validity of the developed program the numerical calculations for Wigley hull and Series 60( <TEX>$C_B=0.6$</TEX>) hull have been performed and the results obtained by the numerical calculations have been compared with the original hulls.