基于等几何分析的结构拓扑优化设计方法研究

Merging the isogeometric analysis (IGA) and structural optimization is an important new direction for the field of computational mechanics, which needs the deepened research. This project aims to establish and develop the optimization model, design method and efficient algorithm of the IGA-based topology optimization design of structures considering the static and dynamic reliability. Firstly, the key formulation and efficient approach of the IGA-based integrated topology and shape optimization of structures are proposed to unify the design process, due to the fact that the existing topology optimization approaches are mainly based on the implicit solution framework, and the emerging explicit approaches have not closely combined CAD and CAE yet. Subsequently, to enhance the computational accuracy and design efficiency, the isogeometric optimization method of simultaneous material and topology design for functionally graded shell structures is developed with achieving the integration of topology and shape optimization, and simultaneous design of material and structure. Finally, the hybridizable discontinuous Galerkin IGA method is proposed through developing the theory of probability density evolution, and the method of stochastic mechanics analysis and algorithm of reliability-based topology optimization of structures with high efficiency and good stability are devised. Moreover, the innovative optimal topology designs of uncertain continuum structures of plates and shells are obtained, and the propagation mechanism of random uncertainty is revealed. This project studies deeply the topology optimization methods of structures combining with the thriving theory of isogeometric analysis, which has important scientific significance and application value for development of industrial digitalization design technology and the relevant program and software.

等几何分析与结构优化的融合是计算力学领域重要的新方向，亟待深入研究。本项目旨在构建与发展基于等几何分析、考虑静动力可靠度的结构拓扑优化设计的优化模型、设计方法和高效算法。首先，鉴于已有的拓扑优化方法主要基于隐式设计框架，新兴的显式方法尚未紧密结合CAD和CAE，提出基于等几何分析的结构拓扑与形状集成优化设计的关键列式和高效方法，形成统一的设计过程。其次，为提高计算精度和设计效率，发展功能梯度壳体结构的材料和拓扑同步等几何优化方法，实现拓扑和形状的集成优化、材料和结构的一体化设计。最后，发展概率密度演化理论，提出混合间断伽辽金等几何方法，建立高效、稳定的结构随机力学分析方法和可靠度拓扑优化算法，获得不确定性板壳连续体拓扑优化创新方案，并揭示随机不确定性传播机理。本项目结合蓬勃发展的等几何分析理论，深入研究结构拓扑优化设计方法，对工业数字化设计技术和程序软件研发具有重要的科学意义和应用价值。

等几何分析紧密联系几何信息，采用相同的数学表达将几何精确建模、结构分析和设计过程结合，为结构优化设计提供了新的选择和机会。相比基于有限元的优化方法，等几何优化设计方法可显著提高结构优化的精度、效率和便利性。瞄准学科前沿和结构优化设计需求，本项目建立了连续体结构应力约束拓扑优化的IGA-SIMP方法，实现了几何建模、结构分析和优化设计的紧密结合。为处理大量局部应力约束同时提高设计优化过程的稳定性，建议了修正的P-norm应力约束结合稳定转换法的策略。分别以局部和全局的方式考虑结构的最大应力约束，提出了两种有效的约束凝聚方案用于求解板壳结构的应力约束拓扑优化问题。而且，构建了基于等几何分析的航空发动机涡轮叶片的形状优化列式，获得了力-热共同作用下叶片壳体结构的形状优化设计方案，结果表明等几何优化方法比基于有限元分析的优化方法具有更高的计算效率和精度。最后，从随机积分的视角另辟蹊径建立并命名了随机系统时间-空间域的概率密度积分方程，提出了结构随机振动和可靠度分析的统一新框架：直接概率积分法，突破了大型复杂结构非线性随机动力学分析这一极具挑战性的技术瓶颈；针对含多设计点的结构静、动力可靠度优化问题，结合概率测度变换，提出了基于直接概率积分法的可靠度优化设计方法。此外，获得了弹性板壳结构平稳/非平稳随机振动响应的精确基准解。本项目推动了数字化设计技术和程序软件的发展，并促进了结构优化方法的工程应用。. 发表国际期刊SCI论文23篇，国内期刊论文4篇；相关成果获2021年国际学术会议ICANDVC-2021的Outstanding paper award。合作组织了10个国内外学术会议和专题研讨会，做大会报告4次和邀请报告10次。培养出站博士后一名，毕业博士4名、硕士6名。获批软件著作权4项，申请国家发明专利3项。