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 引用本文: 王选, 胡平, 祝雪峰, 盖赟栋. 考虑结构自重的基于NURBS插值的3D拓扑描述函数法[J]. 力学学报, 2016, 48(6): 1437-1445.
Wang Xuan, Hu Ping, Zhu Xuefeng, Gai Yundong. TOPOLOGY DESCRIPTION FUNCTION APPROACH USING NURBS INTERPOLATION FOR 3D STRUCTURES WITH SELF-WEIGHT LOADS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1437-1445.
 Citation: Wang Xuan, Hu Ping, Zhu Xuefeng, Gai Yundong. TOPOLOGY DESCRIPTION FUNCTION APPROACH USING NURBS INTERPOLATION FOR 3D STRUCTURES WITH SELF-WEIGHT LOADS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1437-1445.

## TOPOLOGY DESCRIPTION FUNCTION APPROACH USING NURBS INTERPOLATION FOR 3D STRUCTURES WITH SELF-WEIGHT LOADS

• 摘要: 在许多如大坝、桥梁等大型土木工程结构中，结构的自重是初始设计阶段必须考虑的重要载荷之一，因此研究自重载荷作用下的结构拓扑优化设计问题具有十分重要的意义.针对考虑自重载荷作用的拓扑优化问题所面临的主要困难，总结了现有处理考虑自重载荷的拓扑优化问题的三类主要方法；提出一种基于非均匀有理B样条（non-uniform rational B-splines，NURBS）基函数插值的拓扑描述函数方法，基于此方法研究了考虑设计依赖自重载荷作用的2D/3D结构优化设计问题.在列式下，高阶NURBS基函数被同时用于三维NURBS实体片中的几何场、位移场及设计变量场插值，实现了几何模型、分析模型和优化模型的有效统一，确保了位移场及设计变量场的高阶连续性；详细推导了基于NURBS基函数插值的考虑自重载荷作用的三维结构拓扑优化模型及其灵敏度列式，并采用移动渐进线方法（method of moving asymptotes，MMA）进行了优化求解；多个算例验证了方法的有效性和稳定性，结果表明，优化迭代过程稳健，收敛快，能够有效地克服自重载荷作用下连续体结构拓扑优化中经常遇到的低密度区域材料的寄生效应及目标函数的非单调性等问题.

Abstract: The self-weight of the structure is of great importance for large civil engineering structures like dams and bridges, and should be taken into account at the initial design stage. Three main methods to deal with the di culties arisen in optimization problems with self-weight loads are summarized. In this paper, a modified topology description function (TDF) approach using the non-uniform rational B-splines (NURBS) interpolation scheme is introduced for optimal design of 2D/3D continuum structures with design-dependent self-weight loads. In the present approach, the NURBS basis function is applied for the approximation of both the displacement field and the geometry, as well as the interpolation of the design variables. Based on this, the design model and analysis model can be combined closely to realize the computational analysis directly on exact geometry. The model of TDF approach using NURBS interpolation and its sensitivity analysis are detailed. And the method of moving asymptotes (MMA) algorithm is used to solve this optimization problem. Then several numerical examples are performed. It can be seen that the present TDF approach is a robust, fast convergence algorithm, and can effectively overcome the parasitic effect associated with low material density areas, and the nonmonotonous behavior of the compliance that often encountered in topology optimization problems with self-weight loads.

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