• International Journal of CAD/CAM
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• v.7 no.1
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• pp.11-20
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• 2007

A topology optimization methodology, named "smooth boundary topology optimization," is proposed to overcome the shortcomings of cell-based methods. Material boundary is represented by B-spline curves and their control points are considered as design variables. The design is improved by either creating a hole or moving control points. To determine which is more beneficial, a selection criterion is defined. Once determined to create a hole, it is represented by a new B-spline and recognized as a new boundary. Because the proposed method deals with the control points of B-spline as design variables, their total number is much smaller than cell-based methods and it ensures smooth boundaries. Differences between our method and level set method are also discussed. It is shown that our method is a natural way of obtaining smooth boundary topology design effectively combining computer graphics technique and design sensitivity analysis.

• Journal of applied mathematics & informatics
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• v.8 no.3
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• pp.819-827
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• 2001

An algorithmic approach to degree reduction of B-spline curves is presented. The new algorithms are based on the blossoming spline curves are obtained by the generalized least square method. The computations are carried out by minimizing the $L_2$ distamce between the two curves.

• Journal of the Korean Society of Fisheries and Ocean Technology
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• v.27 no.1
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• pp.75-82
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• 1991

The present paper describes a preliminary design method by using the computer graphics for creation of the fore and after body profiles of fishing vessel. It is well known that the Form Parameter design method has some merits at an early stage of design, and the B-spline curve generation technique has some prior properties in representing hull form with the computer graphic. The B-spline curve generation technique combined with the form parameter design method is employed to generate the profiles of fishing vessel. For fore body the stem profiles with bulbous bulb or without one are considered. And for after body the stern profiles of cruiser type and the transom type are generated with stern bulb or with shoe piece. Several examples will shown.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.34 no.8
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• pp.1007-1013
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• 2010

A new local refinement scheme for spline finite element method has been proposed; this scheme involves the use of hierarchical B-spline. NURBS has been widely used in CAD; however, the local refinement of NURBS is difficult due to its tensor-product property. In this study, we attempted to use hierarchical B-splines as local refinement strategy in spline FEM. The regions of high gradients are overlapped by hierarchically-created local meshes. Knot vectors and control points in local meshes are extracted from global meshes, and they are refined using specific schemes. Proper compatibility conditions are imposed between global and local meshes. The effectiveness of the proposed method is verified on the basis of numerical results. Further, it is shown that by using a proposed local refinement scheme, the accuracy of the solution can be improved and it could be higher than that of the solution of a conventional spline FEM with relatively lower degrees of freedom.

• Korean Journal of Computational Design and Engineering
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• v.5 no.3
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• pp.276-284
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• 2000

Usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves in a power form is done by either a knot refinement followed by basis conversions or applying a Taylor expansion on the B-spline curve for each knot span. Presented in this paper is a new algorithm, called a direct expansion algorithm, for the problem. The algorithm first locates the coefficients of all the linear terms that make up the basis functions in a knot span, and then the algorithm directly obtains the power form representation of basis functions by expanding the summation of products of appropriate linear terms. Then, a polynomial segment of a knot span can be easily obtained by the summation of products of the basis functions within the knot span with corresponding control points. Repeating this operation for each knot span, all of the polynomials of the B-spline curve can be transformed into a power form. The algorithm has been applied to both static and dynamic curves. It turns out that the proposed algorithm outperforms the existing algorithms for the conversion for both types of curves. Especially, the proposed algorithm shows significantly fast performance for the dynamic curves.

• Journal of the Korea Computer Graphics Society
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• v.17 no.3
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• pp.33-42
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• 2011

We present a new technique for removing interior knots of parametric B-spline curves. An initial curve is constructed by continuous $L_{\infty}$ approximation proposed by Eck and Hadenfeld. We employ Hausdorff distance to measure the shape difference between the original curve and the initial one. The final curve is obtained by minimizing their Hausdorff distance. We demonstrate the effectiveness of our technique with experimental results on various types of planar and spatial curves.

• Journal of the Society of Naval Architects of Korea
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• v.31 no.3
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• pp.31-37
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• 1994

When a lot of input data are not distributed uniformly n a chord-span direction or when the given shape is complicated, it is very difficult to obtain an inverse matrix which represents the smooth Bi-cubic B-spline surface of the initial shape. To overcome this problem, we suggest image Surface Expansion Method(ISE Method) which is suggested for vertex creation of B-spline curves and surfaces. Its basic concept, convergency and verification are shown. Also B-spline curves and Surfaces represented by ISE Method were compared with those represented by the existing method which is based on the inverse matrix method, the pseudoinverse matrix method and the chord length approximation method for vertex yielding. Ship Hull Forms which have Knuckle, Bulbous Bow, Transom and Stern frame were represented by the ISE Method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.4
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• pp.257-265
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• 2009

In this paper we find an explicit form of upper bound of Hausdorff distance between given cubic spline curve and its quadratic spline approximation. As an application the approximation of offset curve of cubic spline curve is presented using our explicit error analysis. The offset curve of quadratic spline curve is exact rational spline curve of degree six, which is also an approximation of the offset curve of cubic spline curve.

• Journal of Integrative Natural Science
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• v.7 no.2
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• pp.124-129
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• 2014

In this paper, we construct an logarithmic spiral-like curve using curvature-continuous quadratic spline and quadratic rational spline. The quadratic (rational) spline has self-similarity. We present some properties of the quadratic spline. Also using this $G^2$ quadratic spline, an approximation of logarithmic spiral is proposed and error analysis is obtained.

• Journal of the Society of Naval Architects of Korea
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• v.39 no.1
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• pp.8-15
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• 2002

A two-dimensional higher order panel method using B-splines has been developed to overcome the disadvantages of the low order panel method and to obtain more accurate solution. The sources and the normal dipoles are distributed on both the body and the free surface. Instead of applying the upwind finite difference schemes to satisfy the linearized free surface and the radiation condition, the derivatives of the basis functions of the B-splines are directly applied to the linearized free surface condition. Numerical damping in the Dawson's method are avoided in the Present computations. In order to validate the present method, numerical computations are carried out for a submerged cylinder and a two-dimensional hydrofoil steadily moving beneath a free surface. The numerical results show that fast convergence and better accuracies have been achieved by the present method.