材料结构拓扑优化中的数学问题

Structure is man-made load carrying solid. Structural topology optimization aims at finding optimum structural topology, which optimizes its mechanical performances under material volume or economical constraints. Structural topology optimization is the hottest research area in structural optimization in recent 30 years. Since it provides method to change structural topology, select optimum topology, it becomes a design tool for creative design in engineering community and the method and the corresponding software have been developed very fast. Wide application has been found in aerospace and aeronautic industry. And its application has been extended to the disciplinary other than structural engineering, including optimal design of material microstructure and other physical disciplinary. In structural topology optimization, mechanical and other physical quantities defined in two or three dimensional domain are the objective and constraints, selection of optimum topology from various topologies is essentially discrete optimization. To reduce computational burden, many approaches such as SIMP (Solid Isotropic Material with Penalty) and level set method have been proposed to realize continuous transformation between different topologies. For different physical problems and different physical quantities, the connectivity of definition domain and continuity of constraints and objective is different; it creates a number of problems, including continuity of constraints, switch of optimum vibration mode, singular optimum design and symmetry of global optimum. Highly efficient computational algorithms are also highly desired.

结构是人造的承力固体。结构拓扑优化在体积或经济等约束下，寻求力学性能最优的结构拓扑。结构拓扑优化是近三十年来结构优化的热点，由于其提供了改变结构拓扑、选择最优拓扑的方法，为工程界提供了创新设计的工具，因此，这一方法和相应的软件发展非常迅速，在航空航天等工业中得到日益广泛的应用，并且迅速地应用到结构以外的领域，包括材料微结构的优化及其他物理领域。在拓扑优化问题中，定义在二维或三维区域上的力学量及其他物理量是优化问题的目标或约束，从具有不同拓扑的区域中选择最优的拓扑，本质上是离散优化问题。为了降低其计算量，提出了很多算法，如变密度法、水平集法，以实现不同拓扑见的连续变换。对于不同的物理问题和不同的物理量，定义域的连通性和物理量的连续性具有不同特征，由此产生了一系列问题，如约束的连续性、奇异最优解、最优振动或失稳模态的跳跃，还有最优拓扑的对称性；也需要研究提高这些方法的效率和性能的理论和技术。

拓扑优化是近年来在结构优化领域内发展迅速并日益得到广泛应用的一个重要研究方向。结构拓扑中蕴含着大量深刻的数学问题，已经有一批数学工作者进入这一领域开展研究工作。本项目以拓扑优化为背景，致力于为计算力学工作者和应用数学学者围绕拓扑优化这一主题进行跨学科研究提供交流平台、提供文献信息、创造合作机会，推动结构优化这一方向上基于问题驱动的应用数学研究。在本项目支持下，已成功组织了二次专题研讨会。通过邀请国内外在结构拓扑优化领域有重要工作的学者（如国际多学科优化协会现任主席丹麦学者Ole Sigmund，国际理论和应用力学协会前秘书长、丹麦学者Niels Olhoff、韩国工程院院士Kwak Byung Man等）向与会的应用数学/力学领域青年学者介绍拓扑优化这一前沿领域的发展历史、现状及所存在的挑战性问题，促进应用数学工作者与力学工作者协同合作，围绕拓扑优化这一主题开展合作研究。项目执行期间，我们还借助数学方法针对拓扑优化中一些难点问题开展了研究，在拓扑优化特征尺寸控制、拓扑优化计算框架、水平集方法拓展、结构拓扑优化对称性研究、结构可靠度优化问题的数学建模、点阵材料/结构多尺度并发的拓扑优化设计等方面获得了一批成果。部分成果引起了数学工作者的关注。