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

Abstract

Abstract. 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.Key words. Shape optimization, one-level additive Schwarz preconditioner, parallel computing, one-shot method, finite element, inexact Newton.AMS subject classifications. 49K20, 49Q10, 65F08, 65F10, 65N30, 65N55, 65Y05, 76D05, 90C06, 90C30, 93C201. Introduction. Shape optimization, or optimal shape design, aims to optimize an objective function by changing the shape of the computational domain. In the past few years, shape optimization problems have received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis to the problems; see e.g., [19,20,21,27,39,46] and references therein. On the practical side, optimal shape design has played an important role in many industrial applications, for example, aerodynamic shape design [25,33,34,35,44], artery bypass design [2,3,6,7,40,43], microfluidic biochip design [4,24,38] and so on. Most of these optimization problems have constraints imposed by partial differential equations (PDE) or other physical and geometrical conditions. They are considerably more difficult and expensive to solve than the corresponding simulation problems, and often require large scale parallel computers for their ...

Topology optimization study of arterial bypass configurations using the level set method

“…Quarteroni, Rozza and Agoshkov (Quarteroni and Rozza 2003;Rozza 2005;Agoshkov et al 2006) presented a shape optimization method based on the adjoint formulation for the design of aorto-coronaric bypass anastomoses using the steady and unsteady Stokes equations of the Newtonian fluid. Probst et al (Probst et al 2010b) and Chen and Cai (Chen and Cai 2012) applied the gradient-based method for the study of arterial bypass configurations constrained by steady incompressible Navier-Stokes equations of the Newtonian fluid. Abraham et al (Abraham et al 2005a, b) studied the non-Newtonian fluid effects on the optimal design results of arterial bypass grafts considering the steady and unsteady pulsatile blood flows by using a gradient-based optimization algorithm.…”

A parallel two‐level domain decomposition based one‐shot method for shape optimization problems

“…We use a LBBstable Q 2 Q 1 finite element method to discretize problem (24) on a unstructured moving mesh, and a Q 2 finite element method for the moving mesh Equations (5); see Appendix A for details of the discretization. Please find the details of the evaluation of the Jacobian and Hessian in [31,52]. Our algorithms are implemented using PETSc [53].…”

“…The scalability of DDM is wellstudied for scalar elliptic equations [39] and some Schwarz type preconditioning technologies for boundary control problems can be found in [32,38]. Through our numerical experiments [31], we find that the one-level Lagrange-Newton-Krylov-Schwarz method works well for shape optimization problems when the number of processors is small, but when the number of processors is large, unfortunately, the standard two-level method doesn't work as expected because the computational domain on the coarse-level is not exactly the same as the computational domain on the fine-level. In this paper, we introduce a special interpolation method and a new two-level method that works well for shape optimization problems.…”

“…An alternative approach is the simultaneous analysis and design (SAND), or the so-called one-shot method, which solves the three equations in (4) simultaneously by a nonlinear solver, for example, Newton-Krylov methods [6,9,17,19,[28][29][30]. The main challenges of SAND are that the corresponding Jacobian system in the Newton step is large and very ill-conditioned, but, in general, the method 947 based on Newton-Krylov with a good preconditioner is more robust than the method based on nonlinear Gauss-Seidel [31][32][33]. To answer the challenges facing in the one-shot methods, one needs to design a preconditioner that can substantially reduce the condition number of the large fully coupled system and, at the same time, provides the scalability for parallel computing.…”

Two-Level Space–Time Domain Decomposition Methods for Three-Dimensional Unsteady Inverse Source Problems

“…Because of its essential sequential feature, the reduced space SQP method is less ideal for parallel computers with a large number of processor cores, compared with the full space SQP methods. Full space methods were studied for steady state problems in [9,11], but for unsteady problems it needs to eliminate the sequential steps in the outer iteration of the SQP and solve the full space-time system as a coupled system. Because of the much larger size of the system, the full space approach may not be suitable for parallel computer systems with a small number of processor cores, but it has fewer sequential steps and thus offers a much higher degree of parallelism required by large scale supercomputers [12].…”