板的最佳形状设计问题
An optimal shape design problem for plates
摘要：
本文考虑板方程的最优形状设计问题，其中板的变厚度是设计函数。这个问题可以表示为一个控制在系数偏微分方程约束的最优控制问题与额外的控制和状态约束。状态约束处理的Moreau-Yosida正则化的对偶问题。变分离散化用于最优控制问题的离散逼近。对于离散化的状态的混合制剂，我们比较了标准的连续分段线性模型与分段常数的基础上的最低阶Raviart-Thomas混合有限元。我们推导出的离散化和正则化误差的界限，也解决正则化参数和有限元网格大小的耦合。用半光滑牛顿法实现了最优控制问题的数值解。数值算例表明了该方法的有效性。
英文摘要：
We consider an optimal shape design problem for the plate equation, where the variable thickness of the plate is the design function. This problem can be formulated as a control in the coefficient PDE-constrained optimal control problem with additional control and state constraints. The state constraints are treated with a Moreau-Yosida regularization of a dual problem. Variational discretization is employed for discrete approximation of the optimal control problem. For discretization of the state in the mixed formulation we compare the standard continuous piecewise linear ansatz with a piecewise constant one based on the lowest-order Raviart-Thomas mixed finite element. We derive bounds for the discretization and regularization errors and also address the coupling of the regularization parameter and finite element grid size. The numerical solution of the optimal control problem is realized with a semismooth Newton algorithm. Numerical examples show the performance of the method.
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