Relevant integrals of NURBS and its application in hull line element design

NURBS are a powerful mathematical tool for representing free-form curves and surfaces and have been the international industrial standards for the representation, design, data exchange of geometric information. In this paper, starting from the NURBS matrix form, a piecewise-integration is used to derive the formula of relevant integrals of NURBS curve based on the properties of NURBS. A benefit of matrix representation for NURBS is that the explicit mathematical expression of each curve segment is obtained. The area and perimeter of a circle and an ellipse are evaluated from the matrix representation for NURBS. The integrand function from the matrix is solved by composite Newton-Cotes formula and composite Gauss-Legendre formula respectively and the relative error is evaluated with the analytical solution. Results show that three-point composite Gauss-Legendre rule has fast convergence, high accuracy and low evaluation cost. Subsequently, the formula of relevant integrals of NURBS curve is applied to the hull line elements. Taking the hull sectional area curve (SAC) as an example, the evaluation of prismatic coefficient (CP) and longitudinal center of buoyancy (LCB) is carried out; taking the design waterline (DWL) as an example, the evaluation of waterplane coefficient (CW) and transverse center of flotation (Xf) is carried out; taking the middle cross section as an example, the evaluation of midship section area coefficient (CM) is carried out. Finally, taking the revolution as an example, the surface area and volume are evaluated. Through these real ship examples, the effectiveness and universality of the engineering application in hull line element design are verified.

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  • English

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  • Accession Number: 01843268
  • Record Type: Publication
  • Files: TRIS
  • Created Date: Apr 25 2022 10:06AM