舰船科学技术  2024, Vol. 46 Issue (5): 145-148    DOI: 10.3404/j.issn.1672-7649.2024.05.026 PDF

Research on CAD modeling of main hull model based on NURBS surface theory
LI Shao-nan
Henan Provincia1 Research Center of Wisdom Education and Intelligent Technology, Application Engineering Technology, Zhengzhou 450000, China
Abstract: This article studies the theory of NURBS surfaces, with a focus on analyzing the definitions and expressions of NURBS curves and surfaces. It provides a calculation method for B-spline curves, studies the solution method for smoothing ship curves and surfaces, and introduces the definition of the sum of squared shear jumps; Studied the calculation method for three-dimensional stability of ships and the algorithm for solving the static stability force arm of ships, and analyzed the variation curve of the static stability force arm and the variation curve of the ship's draft depth; Finally, the CAD modeling technology for ship hull models was studied, and a design method for a fast ship modeling system was provided.
Key words: surface theory     hull model     modeling
0 引　言

1 NURBS曲面理论 1.1 NURBS曲线曲面的表达

B样条基函数存在多种形式的定义，由于De Boor-Cox-Mansfield基函数在计算机上较容易实现，因此本文采用该类型的基函数递推公式[5]。假设存在非递减数列，如下式：

 ${U^{'}} = \left\{ {{u_0}} \right., \cdots ,\left. {{u_n}} \right\},{u_i} \leqslant {u_n} \text{，}$ (1)

 $\left\{ {\begin{array}{*{20}{l}} {{B_{i,0}} = \left\{ {\begin{split} &{1,}\quad{{u_i} \leqslant u \leqslant {u_{i + 1}}}，\\& {0,}\quad{\rm{others}} ，\end{split}} \right.} \\ {{B_{i,p}}\left( u \right) = \displaystyle\frac{{u - {u_i}}}{{{u_{i + p}} - {u_i}}}{B_{i,p - 1}}\left( u \right) + \displaystyle\frac{{{u_{i + p + 1}} - u}}{{{u_{i + p + 1}} - {u_{i + 1}}}}{B_{i + 1,p - 1}}\left( u \right)} \text{。} \end{array}} \right.$ (2)

 ${C_i}\left( u \right) = \sum\limits_{i = 0}^p {{B_{j,p}}\left( u \right){V_{i + j}}} \text{。}$ (3)

B样条曲线实际应用的过程中，一般为已知测量数据，需构建出一条通过已有测量数据的B样条曲线，即通过给定的测量数据求解出B样条曲线，满足下式：

 $\left\{ {\begin{array}{*{20}{l}} {{V_0} = {P_0}} ，\\ {{V_{n + 2}} = {P_n}} 。\end{array}} \right.$ (4)

B样条曲线受到单参数控制相比，B样条曲面会受到2个参数的控制。假设Vi,j为控制顶点，并且B样条曲面的2个控制参数方向分别为UW，分别如式（5）和式（6）所示，B样条曲面的定义如式（7）所示。

 $U = \left\{ {\underbrace {0,...,0}_{p + 1},{u_{p + 1}},...,{u_n},\underbrace {1,...,1}_{p + 1}} \right\}\text{，}$ (5)
 $W = \left\{ {\underbrace {0,...,0}_{q + 1},{w_{q + 1}},...,{w_m},\underbrace {1,...,1}_{q + 1}} \right\}\text{，}$ (6)
 $S\left( {u,w} \right) = \sum\limits_{i = 0}^n {\sum\limits_{j = 0}^m {{B_{i,p}}\left( u \right){B_{j,q}}\left( w \right){V_{i,j}}} } \text{。}$ (7)

 $C\left( u \right) = \sum\limits_{i = 0}^n {{V_i}{R_{i,p}}\left( u \right)} \text{，}$ (8)

Ri,p(u)的表达式，如式9所示，表示p次有理基函数。

 ${R_{i,p}}\left( u \right) = \frac{{{W_i}{B_{i,p}}\left( u \right)}}{{\sum\limits_{j = 0}^n {{W_j}{B_{j,p}}\left( u \right)} }}\text{。}$ (9)

 $S'\left( {u,w} \right) = \frac{{\sum\limits_{i = 0}^n {\sum\limits_{j = 0}^m {{B_{i,p}}\left( u \right){B_{j,p}}\left( w \right){W_{i,j}}{V_{i,j}}} } }}{{\sum\limits_{i = 0}^n {\sum\limits_{j = 0}^m {{B_{i,p}}\left( u \right){B_{j,q}}\left( w \right){W_{i,j}}} } }}\text{，}$ (10)

 ${W_{i,j}} = \left\{ {{w_0},...,{w_{m + q + 1}}} \right\}\text{。}$ (11)
1.2 船体曲线曲面光顺

 $A = \frac{1}{2}{\left( {EI} \right)^2}\int_0^L {{{\left| {k\left( s \right)} \right|}^2}{\mathrm{d}}s} \text{。}$ (12)

 ${E_c} = \int_0^b {{{\left| {C''\left( u \right)} \right|}^2}{\mathrm{d}}u} \text{。}$ (13)

 ${T_C} = \sum\limits_{l = 0}^{n + 2} {{{\left( {V_l^* - {V_l}} \right)}^2} \lt \varepsilon } \text{。}$ (14)

 ${F_C} = \alpha \cdot \int_a^b {{{\left| {C''\left( u \right)} \right|}^2}du + \sum\limits_{l = 0}^{n + 2} {\gamma {{\left( {V_l^* - {V_l}} \right)}^2}} } \text{。}$ (15)

 $\frac{{\partial {F_C}}}{{\partial V_l^*}} = 0\text{。}$ (16)

 ${D_C} = {\sum\limits_{l = 0}^{n - 2} {\left| {\frac{{{k_{l + 2}} - {k_{l + 1}}}}{{{l_{l + 1}}}} - \frac{{{k_{l + 1}} - {k_l}}}{{{l_l}}}} \right|} ^2}\text{。}$ (17)
2 船舶三维模型稳定性计算

 图 1 船舶排水量随吃水深度变化曲线 Fig. 1 Curve of ship displacement variation with draft depth

 $l = {y_B}\cos \phi + {z_B}\sin \phi - {z_G}\cos \phi \text{。}$ (18)

 图 2 静稳性力臂的变化曲线 Fig. 2 The variation curve of static stability force arm
 $K = \frac{{{l_q}}}{{{l_f}}} \geqslant 1\text{，}$ (19)
 $\Delta \theta = 20\left( {\frac{B}{D} - 2} \right) \cdot \left( {k - 1} \right)\text{。}$ (20)

 图 3 船舶吃水深度随压载舱容积变化曲线 Fig. 3 The curve of ship's draft depth changing with the volume of ballast tanks
3 船舶模型建模技术

 图 4 快速建模系统结构图 Fig. 4 Quick modeling system architecture diagram

 图 5 不同肋位上重心位置的变化曲线 Fig. 5 The variation curve of the center of gravity position on different rib positions

 图 6 船舶模型系统开发平台 Fig. 6 Ship model system development platform
4 结　语