中文摘要：

提出基于力密度方法的统一求解策略，实现NURBS曲线和曲面的精确变形。将控制网格类比为空间索网结构，分别建立外载荷变化最小、外载荷最小、杆阻尼长度最小和节点位移最小等四种表达不同几何和物理意义的最小化模型来驱动变形，并以控制顶点在变形前后的位移为变量，建立统一表达的目标函数方程，便于多个最小化模型组合使用，满足特定的变形需要。由于目标函数的二次性以及约束的线性化处理，优化问题可以通过Lagrange乘子法转化为线性问题进行求解。在计算中引入目标函数快速求解算法，使变形设计可以满足实时交互操作的需要。应用实例表明，该算法可以应用于细节特征设计和反求建模等复杂曲面造型。

英文摘要：

An effective solution for exact shape deformation, dealing with both curves and surfaces in NURBS form, is proposed based on the theory of force density method （FDM）. Using an analogy between the control polyhedron and the bar network, four novel minimization models are constructed, i.e. minimization of the forces variation, minimization of the forces, minimization of the damped bars length, minimization of the nodes displacement, which represent different geometrical and physical properties. All objective functions are described by a uniform formula with the displacement of the control points and then the optimization problem is solved efficiently by using quadratic programming with linear constraints. For efficiency, a fast algorithm for evaluating the objective functions is applied and this makes the deformation suitable for realtime interactive design. Experimental examples show that the proposed method can be widely applied to complicated shape modeling, such as detail feature design, reverse modeling, etc.