论文标题
相位场模型中Dirichlet边界条件的数值优化，并应用于纯物质固化
Numerical Optimization of the Dirichlet Boundary Condition in the Phase Field Model with an Application to Pure Substance Solidification
论文作者
论文摘要
与抛物线PDE的分布式控制相反，目前几乎没有与抛物线PDE的Dirichlet边界条件控制有关的贡献。这激发了我们对相位场模型的Dirichlet边界条件控制的兴趣，该模型描述了超冷熔体对纯物质的凝固。特别是，我们的目的是控制计算域边界上温度场的时间演变，以便在给定时间实现晶体的规定形状。为了获得计算成本功能梯度的有效方法，我们正式得出了伴随问题。然后将梯度用于执行梯度下降。提出的优化方法的生存能力由在一个和两个空间维度中进行的几个数值实验支持。除其他事项外，这些实验表明，在某些情况下，相位场方程中的线性反应项被证明不足，因此考虑了替代反应项来改善模型行为。
As opposed to the distributed control of parabolic PDE's, very few contributions currently exist pertaining to the Dirichlet boundary condition control for parabolic PDE's. This motivates our interest in the Dirichlet boundary condition control for the phase field model describing the solidification of a pure substance from a supercooled melt. In particular, our aim is to control the time evolution of the temperature field on the boundary of the computational domain in order to achieve the prescribed shape of the crystal at the given time. To obtain efficient means of computing the gradient of the cost functional, we derive the adjoint problem formally. The gradient is then used to perform gradient descent. The viability of the proposed optimization method is supported by several numerical experiments performed in one and two spatial dimensions. Among other things, these experiments show that a linear reaction term in the phase field equation proves to be insufficient in certain scenarios and so an alternative reaction term is considered to improve the models behavior.
