Computational Strategies for
       Shape Optimization of Time-Dependent Navier-Stokes Flows
				   
			     Beichang He
		   Engineering Mechanics Laboratory
		       General Electric Company
		    Niskayuna, New York 12309, USA
				   
			     Omar Ghattas
		  Computational Mechanics Laboratory
	  Department of Civil and Environmental Engineering
		      Carnegie Mellon University
		 Pittsburgh, Pennsylvania 15213, USA
				   
			   James F. Antaki
		       Artificial Heart Program
			Department of Surgery
	       University of Pittsburgh Medical Center
		 Pittsburgh, Pennsylvania 15213, USA
				   


We consider the problem of shape optimization of two-dimensional flows
governed by the time-dependent Navier-Stokes equations. For this
problem we propose computational strategies with respect to
optimization method, sensitivity method, and unstructured meshing
scheme. We argue that, despite their superiority for steady
Navier-Stokes flow optimization, reduced sequential quadratic
programming (RSQP) methods are too memory-intensive for the
time-dependent problem. Instead, we advocate a combination of
generalized reduced gradients (for the flow equation constraints) and
SQP (for the remaining inequality constraints). With respect to
sensitivity method, we favor discrete sensitivities, which can be
implemented with little additional storage or work beyond that
required for solution of the flow equations, and thus possess a
distinct advantage over discretized continuous sensitivities, which
require knowledge of the entire time history of the flow variables.
Finally, we take a two-phase approach to unstructured meshing and grid
sensitivities. Far from the optimum, we remesh each new shape
completely using an unstructured mesh generator to accommodate the
large shape changes that are anticipated in this phase, while an
inconsistent but easily computed form of grid sensitivities is
employed. Close to the optimum, where differentiability of the mesh
movement scheme and consistency of grid sensitivities are desirable,
we use elastic mesh movement to generate meshes corresponding to new
shapes. Elastic mesh movement is valid only for small shape changes
but is differentiable and permits computation of exact grid
sensitivities in a straightforward manner. Two examples characterized
by a viscous dissipation objective function illustrate the approach.

BibTex citation:

@techreport{he-ghattas-antaki-97,
author = {Beichang He and Omar Ghattas and James F. Antaki},
title =  {Computational Strategies for Shape Optimization of
          {N}avier-{S}tokes Flows},
number = {CMU-CML-97-102},
institution = {Computational Mechanics Lab},
address = {Department of Civil and Environmental
           Engineering, Carnegie Mellon University},
month = june,
year = {1997}}