Parallel One-Shot Lagrange-Newton-Krylov-Schwarz Algorithms for Shape Optimization of Steady Incompressible Flows

We propose and study a new parallel one-shot Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible NavierStokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems solve iteratively the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally not easy to converge as they are basically nonlinear Gauss-Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization problems in which the mesh equations are implicitly coupled with the optimization problems. Second, we introduce a LNKSz framework based on an overlapping domain decomposition method for solving the fully coupled problem. Such an approach doesn’t involve any sequential steps that are necessary for the Gauss-Seidel type reduced space methods. The main challenges in full space approaches are that the corresponding nonlinear system is much harder to solve because it is two to three times larger and its indefinite Jacobian problems are also much more ill-conditioned. Effective preconditioning becomes the most important component of the method. Numerically, we show that LNKSz deals with these challenges quite well. As an application, we consider the shape optimization of an artery bypass problem in 2D. Numerical experiments show that our algorithms perform well on supercomputers with hundreds of processors.