复合薄膜结构失稳的多层级均匀化建模与仿真

Instability phenomena of composite membranes are widely observed in both nature and engineering. Such phenomena are usually accompanied by large deformation on both material-level and structural-level that are closely related to each other. Direct simulation on these phenomena requires extremely intensive computational cost. Thus, this project aims to develop an accurate and efficient multilevel model by combining the multilevel finite element method (FE2) and the Fourier-based FE model in a complementary way. On the material-level, the three-dimensional Representative Volume Element (RVE) representing the full thickness of the membrane is analyzed to obtain the average stress, so as to accurately provide the real-time constitutive laws to the integration points of structural-level model. On the structural-level, a reduced shell model (i.e., the envelope-level model) is constructed based on a seven-parameter shell model adopting a complete three-dimensional constitutive law without condensation, whose initial unknowns are expanded into Fourier series and replaced by their Fourier coefficients. As the latter varies much more slowly than the initial functions, the reduced shell model could significantly improve the computational efficiency. Then, the FE2 method that is powerful in analyzing complex nonlinear problems is used to realize the real-time information transition among the three levels. To solve the established nonlinear multilevel systems, an accurate yet efficient path following technique, i.e., Asymptotic Numerical Method is used, and a bifurcation indicator is introduced to precisely detect the bifurcation point. In this way, the strong nonlinear path can be quickly traced and the evolution of instability mode can be precisely captured. The outcomes of this project will provide a reliable and efficient technical tool, which will benefit the exploration of key factors of instability and promote the design of advanced composite membrane structures.

复合薄膜结构的失稳是典型的强非线性多层级力学行为。宏观结构层级的失稳与细观材料层级的瞬时应力状态紧密相关、交互影响。本项目旨在开发宏细观并行的高效数值模型，同时寻求其可靠的非线性求解方案，以期快速准确预测复合薄膜失稳并探究其关键影响因素。在细观材料层级上，采用三维代表体元以精细计算在当前宏观结构变形状态下细观应力的平均，从而准确预测宏观模型各积分点的实时本构关系。在宏观结构层级上，基于三维本构的壳体模型，采用傅立叶级数构建单元大小独立于失稳波长的壳体包络线模型，从而大幅减少宏观单元数与积分点数。在层级关联上，在大变形计算均匀化的框架下实现宏、细观模型的并行与信息实时互传。在非线性问题求解上，采用准确高效的数值渐近法，并引入分岔指数以精准定位分岔点，从而快速跟踪宏细观相互纠缠状态下的强非线性路径并甄别失稳模态及其演化。项目研究成果以期为先进复合薄膜结构的优化设计提供理论基础与技术支持。

复合薄膜结构的失稳是典型的强非线性多层级力学行为，寻求高效准确的多尺度建模与仿真方法、稳健可靠的非线性求解方案，对快速准确预测复合薄膜失稳并探究其关键影响因素有重要的科学意义。为此，本项目综合利用傅里叶级数、壳体理论、计算均匀化以及数据驱动算法，构建了可实时跟踪宏、细观力学信息的高效可靠的多尺度均匀化模型。另外，发展了基于高阶幂级数的数值渐近法，并引入基于稳定性理论的分岔指数，实现了失稳临界点的快速精准定位及后屈曲行为的可靠预测。基于上述多尺度均匀化计算方案，探究了复合薄壳结构多稳态行为的关键影响参数，发现了与薄膜结构失稳相关的关键无量纲参数。此外，项目围绕高质量数据库构建方法及高效数据驱动算法，进一步开展了数据驱动计算力学方面的研究工作。研究结果显示，本项目构建的多尺度建模与仿真方案为快速准确预测复合薄膜失稳及探究其关键影响因素提供了可靠的理论基础与技术支持。此外，计算均匀化理论、模型缩减技术以及数据驱动算法的融合对于拓宽力学与数据科学的交叉前沿具有重要的科学意义。依托本项目，项目组在JMPS、CMAME、IJSS等本领域重要学术期刊上发表期刊论文12篇，项目负责人在第15届世界计算力学大会（WCCM2022）上作特邀报告（Keynote lecture）。联合培养高素质研究生5名，其中黄威获第23届国际复合材料与结构会议“Ian Marshall 最佳学生论文奖”（首位获奖华人）及“武汉大学学术创新奖一等奖”，匡增涛获“武汉大学学术创新奖二等奖”。