The Research Based On Scalable Parallel Methods For Unsteady Incompressible Flows

PDE-constrained optimization problem refers to the optimization of systemsgoverned by partial diferential equations (PDEs). The simulation problem is tosolve the PDEs for the state variables (e.g. displacement, velocity, temperature,electric feld, magnetic feld, species concentration), given appropriate data (e.g.geometry, coefcients, boundary conditions, initial conditions, source functions).The optimization problem seeks to determine some of these data the decisionvariables given performance goals in the form of an objective function and pos-sibly inequality or equality constraints on the behavior of the system. Since thebehavior of the system is modeled by the PDEs, they appear as (usually equal-ity) constraints in the optimization problem. The size, complexity, and infnite-dimensional nature of PDE-constrained optimization problems present signifcantchallenges for general-purpose optimization algorithms. These features often re-quire regularization, iterative solvers, preconditioning, globalization, inexactness,and parallel implementation that are tailored to the structure of the underlyingoperators.As the technology and size of parallel computing systems advance, so does thedetails at which we can solve very complex numerical problems, including optimiza-tion problems constrained by nonlinear partial diferential equations (PDEs). Thistrend of increasing computational complexity demands both the design of scalableparallel algorithms and the adoption of modern software engineering techniquesfor the developments of numerical libraries. This thesis presents the developmentthe robust parallel numerical methods for solving PDE-constrained optimizationproblems and the corresponding simulation problem. We proposed a new class offully coupled full space parallel algorithms. This thesis consists of the followingparts.In Chapter2, we investigate some fully coupled parallel Newton-Krylov-Schwarz (NKS) algorithms for the simulation problem of unsteady incompress-ible flows governed by the Navier-Stokes equations. The algorithms include twomajor parts: a nonlinear Newton method for the outer iteration and a two-levelSchwarz preconditioner for the linear part of the problem. Most of the existingapproaches for this kind of simulation problems are based on the so-called reducedspace method which is easier to implement but may have convergence issues insome situations. In the full space approach we couple the state variables in a sin- gle large system of nonlinear equations. The coupled system is considerably moreill-conditioned than its sub-systems, however, with the powerful NKS approach, weare able to solve these difcult systems efciently on large scale parallel comput-ers. We show numerically that such an approach is scalable in the sense that thenumber of Newton iterations and the number of linear iterations are both nearlyindependent of the grid size, the number of processors, and the Reynolds numbers.We present numerical experiments obtained on supercomputers with more thantwo thousand processors.In Chapter3, We develop a parallel fully implicit domain decomposition algo-rithm for solving optimization problems constrained by time dependent nonlinearpartial diferential equations. In particular, we study the boundary control of un-steady incompressible Navier-Stokes equations. After an implicit discretization intime, a fully coupled sparse nonlinear optimization problem needs to be solved ateach time step. The class of full space Lagrange-Newton-Krylov-Schwarz (LNKS)algorithms is used to solve the sequence of optimization problems. Among op-timization algorithms, the fully implicit full space approach is considered to bethe easiest to formulate and the hardest to solve. We show that LNKS, with arestricted additive Schwarz preconditioner, is an efcient class of methods for solv-ing these hard problems. To demonstrate the scalability and robustness of thealgorithm, we consider several problems with a wide range of Reynolds numbersand time step sizes, and we present numerical results for large scale calculationsinvolving several millions unknowns obtained on machines with more than twothousand processors.In Chapter4, we present some parallel semismooth Newton-Krylov-Schwarz(NKS) algorithm for solving a kind of optimization problems constrained by in-equality nonlinear partial diferential equations: complementarity problems. Thefamily of Semismooth-Newton-Krylov-Schwarz methods is based on a semismoothinexact Newton method, Krylov subspace method, and overlapping Schwarz pre-conditioner. Using semismooth function, the solution of the optimization problemcan be obtained by solving a large sparse nonlinear system of algebraic equations.Numerical results show that the efciency can be achieved by the proposed method.Finally, in Chapter5, we simulate the flow feld distribution within the torqueconverter based on Newton-Krylov-Schwarz algorithms.Our algorithms are implemented based on the Portable Extensible Toolkitfor Scientifc computation (PETSc). In Appendix A, we describe the high level use of PETSc, which is the object-oriented software we have developed for theimplementation of the parallel algorithms based one the proposed optimizationproblem and for the execuation of our parallel numerical experiments.