An inverse problem formulation of the immersed‐boundary method
An inverse problem formulation of the immersed‐boundary method
复制标题
浸没边界法的反问题表述
DOI：
10.1002/fld.4816
复制
发表时间：
2020
影响因子：
1.8
通讯作者：
Hicken, Jason E.
中科院分区：
文献类型：
--
作者：
Yan, Jianfeng;Hicken, Jason E.
我们将浸入边界法（IBM）表述为反问题。在包含目标域的较大域的边界上引入控制变量。最优控制是最小化状态与沿浸没目标域边界沿着的期望边界值之间的失配的控制。我们开始通过研究一个我们发现不适定的简单问题公式：在拉普拉斯方程的情况下，我们证明了解是唯一的，但它不能连续依赖于数据;对于线性平流方程，甚至解的唯一性也不成立。这些问题通过两个互补的战略来解决。第一个策略是确保封闭域趋于真实域，因为网格被细化。第二个策略是包含一个专门的无参数正则化，该正则化基于惩罚边界上的控制和状态之间的差异。提出的逆IBM应用于扩散，对流，对流扩散方程使用高阶间断Galerkin离散。数值实验表明，正则化的计划达到最佳的收敛速度和减少Hessian的优化问题有界的条件数，作为网格细化。
We formulate the immersed‐boundary method (IBM) as an inverse problem. A control variable is introduced on the boundary of a larger domain that encompasses the target domain. The optimal control is the one that minimizes the mismatch between the state and the desired boundary value along the immersed target‐domain boundary. We begin by investigating a naïve problem formulation that we show is ill‐posed: in the case of the Laplace equation, we prove that the solution is unique, but it fails to depend continuously on the data; for the linear advection equation, even solution uniqueness fails to hold. These issues are addressed by two complimentary strategies. The first strategy is to ensure that the enclosing domain tends to the true domain, as the mesh is refined. The second strategy is to include a specialized parameter‐free regularization that is based on penalizing the difference between the control and the state on the boundary. The proposed inverse IBM is applied to the diffusion, advection, and advection‐diffusion equations using a high‐order discontinuous Galerkin discretization. The numerical experiments demonstrate that the regularized scheme achieves optimal rates of convergence and that the reduced Hessian of the optimization problem has a bounded condition number, as the mesh is refined.
椭圆偏微分方程最优控制问题的有限元误差估计
DOI：
--
发表时间：
2009
期刊：
Large-Scale Scientific Computing
影响因子：
--
作者：
F. Tröltzsch
通讯作者：
F. Tröltzsch
作为反问题的浸入边界法
DOI：
10.2514/6.2018-4162
发表时间：
2018
期刊：
2018 AIAA Fluid Dynamics Conference
影响因子：
--
作者：
Yan, Jianfeng;Hicken, Jason E.
通讯作者：
Hicken, Jason E.
DOI：
--
发表时间：
2008
期刊：
Elsevier 93
影响因子：
--
作者：
T.;Suzuki;et. al.
通讯作者：
et. al.