基于等几何的板壳结构几何非线性拓扑优化

Geometric nonlinear topology optimization of plate and shell structures based on isogeometric analysis

摘要

面向板壳结构的补强设计，提出了一种基于等几何分析的几何非线性拓扑优化方法。板壳结构的几何通过NURBS曲面表示，构建了基于退化实体壳单元纵向积分规则的分层模型（蒙皮和补强设计域）。补强设计域采用三场SIMP节点密度插值模型开展拓扑优化寻找最优构型，拓扑优化过程考虑板壳结构的几何非线性，建立了考虑大转动的样条退化壳单元格式并推导了全解析灵敏度。基于壳单元的分层模型相比于传统的实体单元模型大幅降低了模型规模，等几何分析实现了板壳结构拓扑优化与CAD系统的无缝集成。数值算例表明该方法能够在考虑几何非线性影响下获得板壳结构清晰的补强区域。同时，本研究讨论了小变形拓扑优化与几何非线性拓扑优化的差异，验证了板壳结构拓扑优化考虑几何非线性的必要性。

Abstract

A geometric nonlinear topology optimization method based on isogeometric analysis is proposed for the reinforcement design of plate-shell structures. The geometry of the plate-shell structure is represented by NURBS surfaces, and a layered model (skin and reinforcement design domain) is constructed based on the longitudinal integral rules of the degenerate solid shell elements. In the design domain, a three-field SIMP node density interpolation model is used for topology optimization to find the optimal configuration. The geometric nonlinearity of plate-shell structure is considered in the topology optimization process, and the spline degenerate shell element scheme considering large rotation is established and the full analytical sensitivity is derived. The hierarchical model based on shell element greatly reduces the model scale compared with the traditional solid element model, and the geometric analysis realizes the seamless integration between the topology optimization of shell structure and CAD system. Numerical examples show that the method can obtain the clear reinforcement region of the plate-shell structure under the consideration of geometric nonlinearity. At the same time, the differences between small deformation topology optimization and geometric nonlinear topology optimization are discussed, and the necessity of considering geometric nonlinearity in the topology optimization of plate and shell structures is verified.

关键词

等几何分析 / 几何非线性 / 拓扑优化 / 板壳结构

Key words

isogeometric analysis / geometric nonlinearity / topology optimization / plate and shell structures

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基金资助

国家自然科学基金资助项目(11772078)