Efficient Multigrid Isogeometric Topology Optimization under Bézier Element Stiffness Mapping

doi:10.3969/j.issn.1004-132X.2022.23.004

Abstract

Abstract: Isogeometric topology optimization（ITO） utilized B-spline or NURBS as the shape function of unknow physical field of CAE， which avoided the low-accuracy problems of traditional FEM-based topology optimization due to the C0 continuity of Lagrange basis function. Through the inheritance between design variables on different hierarchies， the calculation efficiency of ITO might be significantly improved， while it still suffered from the problems of huge memory burden and complicated pre-processing processes with respect to the elemental stiffness matrices of all elements. To resolve these problems， the multigrid ITO method was put forward based on Bézier element stiffness mapping herein. The standard Bézier element stiffness matrix and the Bézier extraction matrices of all hierarchies were used to express the elemental stiffness matrix of an arbitrary B-spline element on arbitrary grid-level equivalently， which optimized the data storage structure and pre-processing processes for multigrid ITO. Numerical example results show that compared with conventional multigrid ITO method， the proposed method obtains identical optimization processes and results， but significantly reduces the storage burden and pre-processing time of the B-spline elements stiffness matrices. Therefore， the effectiveness of the proposed method was verified.

Key words: , topology optimization, isogeometric analysis, multigrid, Bézier extraction

摘要： 等几何拓扑优化方法采用CAD的B样条或者NURBS作为CAE未知物理场的形函数，有效避免了传统有限元拓扑优化由于拉格朗日基函数C0连续所带来的低精度问题。多重网格等几何拓扑优化技术可通过优化域的层间继承显著提高等几何拓扑优化的计算效率，但存在单元刚度矩阵消耗内存空间及预处理时间过长等问题。针对上述问题，研究了基于Bézier单元刚度映射的多重网格等几何拓扑优化方法，采用标准Bézier单元刚度矩阵与相应的Bézier提取矩阵进行任一层级任一B样条单元刚度矩阵的等效表达，进而实现多重网格等几何拓扑优化数据存储结构和预处理过程的优化。数值算例结果表明，相比于传统多重网格等几何拓扑优化模型，所提方法具有相同的优化收敛过程和优化结果，并显著地减小B样条单元刚度矩阵的存储空间并缩短预处理时间，验证了所提方法的有效性。

关键词: 拓扑优化, 等几何分析, 多重网格, Bézier提取

CLC Number:

TH122

DING Yandong, LUO Nianmeng, YANG Aodi, WANG Shuting, ZHU Haoran, XIE Xianda. Efficient Multigrid Isogeometric Topology Optimization under Bézier Element Stiffness Mapping[J]. China Mechanical Engineering, 2022, 33(23): 2801-2810.

丁延冬, 罗年猛, 杨奥迪, 王书亭, 朱浩然, 谢贤达. Bézier单元刚度映射下的高效多重网格等几何拓扑优化方法[J]. 中国机械工程, 2022, 33(23): 2801-2810.

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