Identification of Overpressure Sources at Launch Vehicle Liftoff Using an Inverse Method

Abstract

A = source operator projection A = operator projection A for Dirac source excited at time ti B = incident field operator projection B = incident field operator B projection for Dirac source excited at time ti c = sound speed in the air Dok = delta of pressures computed for the kth couple of sensors Domesk = delta of pressures measured for the kth couple of sensors e = Dirac source excited at time ti operator H = Hessian approximation it = iteration j p J O p = cost function L = Lagrangian l = iteration optimizer N = no. of solver time iterations NDF = no. of degrees of freedom on the obstacle boundary Ncoupl = no. of diametrically opposed pressure sensors Nparam = no. of source parameters Nobs = no. of pressure sensors n = discrete current iteration O p x; t = total pressure field computed Oi = pressure at the ith receiver Omes x; t = measured observed pressure at sensors Odiff x; t = scattered field outside the obstacle Odiff x; t = scattered field inside the obstacle Opinc x; t = incident field for source p O = adjoint total field O i = adjoint field at the ith sensors p = source parameter vector pi = component of excitation parameter at time t ti Q k = scattered matrices adjoint posttreatment R = integral surface operator R k = surface integral adjoint matrices S p = boundary traces of the incident field source parameter p t = time variable U = acoustic pressure jump across the boundary U = adjoint pressure jump V = unitary jump pressure for Dirac source excited at time ti W = unitary pressure for Dirac source excited at time ti x, y = spatial variables in 3-D x0 = location of source label p = obstacle boundary t = time step = test function (in variational formulation) e = exterior domain obstacle i = interior domain obstacle rj rpi = unitary gradient for Dirac source excited at time ti Received 6 December 2005; accepted for publication 26 May 2006. Copyright ©2006 by theAmerican Institute ofAeronautics andAstronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0022-4650/07 $10.00 in correspondence with the CCC. Expert in Structural Analysis, Missile Structural Analysis Department. Research Development Engineer, Simulation Systems and Information Technology Department. Manager of Applied Mathematics and High Performance Computing Team, Simulation Systems and Information Technology Department. JOURNAL OF SPACECRAFT AND ROCKETS Vol. 44, No. 3, May–June 2007