The immersed boundary method as an inverse problem

Abstract

problem formulation is shown to be ill-posed. The ill-posedness is 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, advection-diffusion, and linear elasticity equations using a high-order discontinuous Galerkin (DG) 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.

The author's contribution to the multidimensional summation-by-parts (SBP) discretization of second-order differential operators is also presented. Specifically, a general SBP discretization is presented that is both adjoint consistent and energy stable. It is shown that the discretization generalizes some well-known DG schemes, including the modified scheme of Bassi and Rebay (BR2), the symmetric interior penalty Galerkin (SIPG) scheme, and the compact discontinuous Galerkin (CDG) scheme. It is shown that SIPG can be obtained from BR2 through matrix analysis, and that the original CDG is not energy stable.

The immersed-boundary method (IBM) is formulated 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. A na\"ive