特征值优化问题的理论和算法研究

项目名称： 特征值优化问题的理论和算法研究

项目编号： No.11201106

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 张郑芳

作者单位： 杭州电子科技大学

项目金额： 22万元

中文摘要： 弹性材料设计和光子晶体带隙结构优化均涉及特征值优化问题求解。从数学模型上来说，两类问题化简后均出现两类特征值模型问题：一类是带分片常数介电函数的负拉普拉斯算子的特征值问题；另一类是带分片常数密度函数的负拉普拉斯算子的特征值问题。本项目从模型问题出发，首先克服不同材料几何形状、拓扑结构事先未知的困难，提出变异水平集方法和形状微分方法，推导特征值优化目标泛函对变异水平集函数的第一变分以及形状导数。其次，针对质量、面积等约束条件提出优化方法，转化为无约束问题。第三，对多重特征值优化问题，建立新的目标泛函，以克服特征值为多重时梯度型方法失效的局限性。最后，把模型问题的分析方法和数值算法推广到两类实际问题中，实现加筋设计问题、梁的设计问题、悬臂的设计问题以及光子晶体的带隙结构优化问题在二维、三维中的数值模拟计算。

中文关键词： 特征向量；特征值；优化；反问题；非线性

英文摘要： The design of elastic materials distribution and bandgap optimization of photonic crystal are both modeled by the eigenvalue optimization problems. That is to say, both practical problems could be reduced into two simpler model problems: the eigenvalue of -Δ with piecewise constant dielectric function and the one with piecewise constant density function. First, since the shape and topology of the domains with different materials is unkown as a prior, it is difficult to deal with the problems. To overcome this, we propose alternative piecewise constant level set methods and shape derivative approach. In detail, we will deduce both the first variation and shape derivative of the eigenvalue functional. Second,we propose the constraint optimization method to deal with the area and mass constraints. Third, to overcome the eigenvalue multiplicities, new eigenvalue functional will be set. Finally, we apply the analytical and numerical approaches of the model eigenvalue optimization problems into the elastic materials distribution problems and bandgap optimization structure of photonic crystal. The numerical simulation in 2D and 3D will be conducted, including reinforcement optimization design、beam layout optimization, the shape optimization of the eigenvalue of a cantilever and bandgap optimization structure of photo

英文关键词： eigenmode；eigenvalue；optimization；inverse problem；nonlinear