Shape optimization of isogeometric boundary element method using flower pollination algorithm
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摘要:
为了避免离散整个结构，以及减少几何模型与离散模型之间的误差，提出一种改进的花朵授粉算法优化等几何边界元模型的形状优化算法。采用的边界元法仅需离散结构边界，简化了离散过程，且加入的等几何分析可提高计算精度。以最小化位移或应力为目标函数，以结构面积等于指定面积为约束条件，采用增广乘子法将约束优化模型转化为无约束优化模型；通过等几何边界元法对结构的受力情况进行分析；采用精英反向学习策略及大规模分布估计算法（LSEDA）改进花朵授粉算法，通过花朵授粉算法优化控制点的坐标值，并通过非均匀有理 B 样条（NURBS）基函数构建结构的边界，输出最优结构的形状。Ackley函数的测试结果表明：改进的花朵授粉算法14步收敛，而原始花朵授粉算法136步收敛，且所得最小值为8.881 8×10−16，小于0.001 4，改进的花朵授粉算法寻优能力更强。形状优化的计算结果表明：所提算法可有效求解二维等几何边界元形状优化问题。
Abstract:In order to avoid discretizing the whole structure and reduce the error between the geometric model and the discrete model, a shape optimization method based on the flower pollination algorithm and isogeometric boundary element method is proposed. The boundary element method only needs to discretize the structure boundary, and this simplifies the discretization process. The boundary element method only needs to discretize the structure boundary, and this simplifies the discretization process. Furthermore, the isogeometric analysis is exploited to improve the accuracy. Firstly, the objective function is to minimize the displacement or stress, and the constraint condition is that the structural area is equal to the specified area; secondly, the constrained optimization model is transformed into an unconstrained optimization model by using the augmented lagrange multiplier method; Then, the bearing condition of the structure is analyzed by the isogeometric boundary elementmethod (IGABEM); finally, the elite opposite-based learning strategy and large scale estimation of distribution algorithms (LSEDA) are used to improve the flower pollination algorithm, and the coordinates of control points are optimized by the improved flower pollination algorithm. After that, the boundary of structure is constructed by the non-uniform rational B-splines (NURBS) basis function, and the optimal structure shape is drawn. According to the test findings for the Ackley function, the enhanced flower pollination algorithm converges in 14 steps as opposed to 136 steps for the original flower pollination algorithm, and its minimum value is 8.881 8×10−16 less than 0.0014, demonstrating that it is more capable of optimization. The computational results of shape optimization show that the proposed algorithm can effectively solve two-dimensional shape optimization problem based on isogeometric boundary element method.
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表 1 位移算例的控制点及节点向量
Table 1. Control points and knot vectors for displacement example
参数 数值 ${\boldsymbol{P}}$ [0, 0; 10, 0; 20, 0; 20, 5; 20, 10; 20, 15; 20, 20;
10, 20; 0, 20; 0, 15; 0, 10; 0, 5; 0, 0]${\boldsymbol{\varXi } }$
[0, 0, 0, 1/6, 1/6, 2/6, 2/6, 3/6, 3/6, 4/6, 4/6, 5/6, 5/6, 1, 1, 1]
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表 2 应力算力的控制点及节点向量
Table 2. Control points and knot vectors for stress example
参数 数值 ${\boldsymbol{P}}$ [0, 0; 10, 0; 20, 0; 20, 5; 20, 10; 20, 15; 20, 20;
10, 20; 0, 20; 0, 15; 0, 10; 0, 5; 0, 0]${\boldsymbol{\varXi } }$
[0, 0, 0, 1/8, 1/8, 2/8, 3/8, 4/8, 4/8, 5/8, 5/8, 6/8, 7/8, 1, 1, 1]
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