A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimisation algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterised boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.
翻译:展示了一套使用跟踪型成本功能解决的二维Stefan问题优化实例。使用了一个水平设定方法来捕捉液相和固相之间的界面,并采用浸入边界(截断单元格)方法以及隐含时间推进办法来解决热等式。然后使用一个保守的隐含的表达式来解决水平设定运输方程式。由此产生的数字框架在Stefan问题的现有分析解决方案方面得到验证。然后使用基于联合的算法来有效计算优化算法(L-BFGS)中使用的梯度。算法遵循一个连续的连接框架,在这个框架中,通过形状的微积分和运输定理器正式生成了双轨方程式。提出了一系列广泛的控制目标,结果显示,使用参数化边界定律使有效的控制策略得以抑制内部不稳定性或保持理想的晶体形状。