Utilizing parallel optimization in computational fluid dynamics

Utilizing Parallel Optimization in Computational Fluid Dynamics by Michael Kokkolaras General problems of interest in computational uid dynamics are investigated by means of optimization. Speci cally, in the rst part of the dissertation, a method of optimal incremental function approximation is developed for the adaptive solution of di erential equations. Various concepts and ideas utilized by numerical techniques employed in computational mechanics and arti cial neural networks (e.g. function approximation and error minimization, variational principles and weighted residuals, and adaptive grid optimization) are combined to formulate the proposed method. The basis functions and associated coe cients of a series expansion, representing the solution, are optimally selected by a parallel direct search technique at each step of the algorithm according to appropriate criteria; the solution is built sequentially. In this manner, the proposed method is adaptive in nature, although a grid is neither built nor adapted in the traditional sense using a-posteriori error estimates. Variational principles are utilized for the de nition of the objective function to be extremized in the associated optimization problems, ensuring that the problem is well-posed. Complicated data structures and expensive remeshing algorithms and systems solvers are avoided. Computational e ciency is increased by using low-order basis functions and concurrent computing. Numerical results and convergence rates are reported for a range of steady-state problems, including linear and nonlinear di erential equations associated with general boundary conditions, and illustrate the potential of the proposed method. Fluid dynamics applications are emphasized. Conclusions are drawn by discussing the method's limitations, advantages, and possible extensions. The second part of the dissertation is concerned with the optimization of the viscous-inviscid-interaction (VII) mechanism in an airfoil ow analysis code. The VII mechanism is based on the concept of a transpiration velocity boundary condition, whose convergence to steady state is accelerated. The number of variables in the associated optimization problem is reduced by means of function approximation concepts to ensure high number of parallel processors to number of necessary function evaluations ratio. Numerical results are presented for the NACA-0012 and the supercritical RAE-2822 airfoils subject to transonic ow conditions using a parallel direct search technique. They exhibit a satisfactory level of accuracy. Speed-up depends on the number of available computational units and increases for more challenging ow conditions and airfoil geometries. The enhanced code constitutes a useful tool for airfoil ow analysis and design and an acceptable alternative to computationally expensive high delity codes. Acknowledgments I am immensely grateful to my supervising advisor, Dr. Andrew J. Meade, Jr., for his great guidance and encouragement. Working with him is a pleasure, especially because of his kind personality. I am deeply indebted to Dr. John Emory Dennis, Jr., for his keen interest and support, and for serving in my dissertation committee. I am profoundly thankful to Dr. Pol D. Spanos, for his precious advice and suggestions, and for his inspiring geniality. I appreciate the helpful comments of Dr. S. Scott Collis. I am obliged to Dr. Boris A. Zeldin for numerous discussions, and to David Sera ni for his continuously prompt assistance with the optimal utilization of the PDS software. This work was made possible through the nancial support under the NASA grant # CRA2-35504. The access to the computing facilities of the Center for Research on Parallel Computation (CRPC) at Rice University, the technical support of system administrator Michael Pearlman, and the short but e ective use of the IBMSP supercomputer of the Texas Center for Computational and Information Sciences (TCCIS) at the University of Houston are highly appreciated. I would also like to thank my colleagues and the sta of the department for providing a pleasant and stimulating working environment. Finally, I deliberately do not acknowledge here the beloved, special persons who have contributed, as to every aspect of my life, to this dissertation, in the hope that my actions have been, are, and will be expressing my love and gratitude towards them.