Sequential Quadratic Programming (SQP) for optimal control in direct numerical simulation of turbulent flow

• Published in: Journal of computational physics Vol. 256; pp. 1 - 16
• Main Authors: Badreddine, Hassan, Vandewalle, Stefan, Meyers, Johan
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2014
• Subjects: Adjoint equations; Algorithms; Computational fluid dynamics; Computer simulation; Damped limited-memory BFGS; Direct Numerical Simulations; Mathematical analysis; Mathematical models; Optimal control; Optimization; Sequential Quadratic Programming; Turbulence; Turbulent flow; Turbulent mixing layer; Turbulent mixing layer; Direct Numerical Simulations; Adjoint equations; Damped limited-memory BFGS; Optimal control; Sequential Quadratic Programming
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2013.08.044

The current work focuses on the development and application of an efficient algorithm for optimization of three-dimensional turbulent flows, simulated using Direct Numerical Simulation (DNS) or Large-Eddy Simulations, and further characterized by large-dimensional optimization-parameter spaces. The optimization algorithm is based on Sequential Quadratic Programming (SQP) in combination with a damped formulation of the limited-memory BFGS method. The latter is suitable for solving large-scale constrained optimization problems whose Hessian matrices cannot be computed and stored at a reasonable cost. We combine the algorithm with a line-search merit function based on an L1-norm to enforce the convergence from any remote point. It is first shown that the proposed form of the damped L-BFGS algorithm is suitable for solving equality constrained Rosenbrock type functions. Then, we apply the algorithm to an optimal-control test problem that consists of finding the optimal initial perturbations to a turbulent temporal mixing layer such that mixing is improved at the end of a simulation time horizon T. The controls are further subject to a non-linear equality constraint on the total control energy. DNSs are used to resolve all turbulent scales of motion, and a continuous adjoint formulation is employed to calculate the gradient of the cost functionals. We compare the convergence speed of the SQP L-BFGS algorithm to a conventional non-linear conjugate-gradient method (i.e. the current standard in DNS-based optimal control), and find that the SQP algorithm is more than an order of magnitude faster than the conjugate-gradient method.