考虑材料分布不确定性的结构拓扑优化问题数学建模与求解方法

This project is motivated by research background of structural configuration design under random fields of material property dispersion and manufacturing error. In the framework of level set-based topology and shape representation, characterization, discretization and dimension reduction methods will be developed for spatially distributed uncertainties associated with material properties, boundary shape and material interfaces. The mathematical formulation for the corresponding topology optimization problems will be constructed. The optimization problem will be solved with mathematical programing algorithms. Through research work, major advances will be achieved in key scientific problems with theoretical and practical significance, on theories and methods of continuum topology optimization considering spatially distributed uncertainties. Novel research achievements will include: (1) Discretization and dimension reduction approximation methods for uncertainties of material distribution, boundary shape and multi-material interfaces in topology optimization problems. (2) Solvable optimization formulation and solution methods for continuum structural topology optimization problems considering material spatial distribution uncertainties. (3) Efficient shape sensitivity analysis scheme of statistics of objective functions under boundary shape perturbation. These results will provide uncertainty models, mathematical formulations and efficient numerical methods for structural topology optimization problems considering spatially distributed uncertainties of material, boundary shape and multi-material interfaces. These result will extend frontier theories and methods of structural topology optimization and structural optimization with uncertainties.

本项目以考虑材料性能分散、制造误差等随机场的结构构型设计问题为背景，在水平集拓扑和形状描述的框架下，发展材料性能、边界形状、材料界面等空间分布不确定性的表征、离散和降维方法，建立相应的拓扑优化问题数学列式，实现其数学规划求解。通过拓扑和形状不确定性的描述模型、优化列式、数值方法和试验验证等工作，将在具有重要理论和应用背景的关键科学问题的研究中取得突破，完成以下创新性成果：（1）拓扑优化中材料空间分布、边界形状、多相材料界面等不确定性的离散和降维近似方法；（2）考虑材料空间分布不确定性的连续体结构拓扑优化问题的可解性优化列式及其求解算法；（3）边界形状扰动下优化目标函数统计特征的形状灵敏度分析高效算法。研究成果将为考虑材料、边界形状和多相材料界面空间分布不确定性的结构拓扑优化问题提供不确定性建模方法、数学列式和高效求解算法，从而丰富和发展结构拓扑优化、不确定性结构优化的前沿理论与方法。

考虑不确定性的结构优化理论与算法是计算力学当前的重要前沿方向之一。本项目以考虑材料性能分散、制造误差等随机场的结构构型设计问题为背景，在水平集拓扑和形状描述的框架下，发展了材料性能、边界形状、材料界面等空间分布不确定性的表征方法，建立了相应的拓扑优化问题数学列式，实现了其数学规划求解。通过拓扑和形状不确定性的描述模型、优化列式、数值方法等工作。. 本项目在具有重要理论和应用背景的关键科学问题的研究中取得了突破，完成了以下创新性成果：（1）拓扑优化中材料空间分布、边界形状、多相材料界面等不确定性的离散和降维近似方法；（2）考虑材料空间分布不确定性的连续体结构拓扑优化问题的可解性优化列式及其求解算法；（3）边界形状扰动下优化目标函数统计特征的形状灵敏度分析高效算法。研究成果为考虑材料、边界形状和多相材料界面空间分布不确定性的结构拓扑优化问题提供不确定性建模方法、数学列式和高效求解算法，从而丰富和发展结构拓扑优化、不确定性结构优化的前沿理论与方法。. 2016年以来，本项目研究工作在计算力学领域顶级期刊Computer Methods in Applied Mechanics and Engineering、International Journal for Numerical Methods in Engineering、结构优化权威期刊Structural and Multidisciplinary Optimization等重要国际期刊发表或接受标注资助号的SCI期刊论文4篇。在第12届世界计算力学大会（The 12th World Congress on Computational Mechanics）等国际学术会议做邀请或Keynote报告3次。亢战2016年入选科技部中青年科技创新领军人才、长江学者特聘教授。

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