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Inverse fluid flow problems with discontinuities

Presented by: 
O Pironneau [Paris VI]
Date: 
Tuesday 21st January 2003 - 09:00 to 10:00
Venue: 
INI Seminar Room 1
Session Title: 
Mathematical Challenges in Scientific and Engineering Computation
Abstract: 
There are many fluid flow problems with discontinuities in the data or in the flow. Among them three are quite important for applications:
  • flow through porous media with several geological layers,
  • transonic and supersonic flow with shocks,
  • accoustics with sonic boom,

Optimisation of these systems by standard gradient methods require the application of the techniques of the Calcul of Variations and an implicit asumption that a Taylor expansion exists with respect to the degrees of freedom of the problem. Take for example the flow in a transonic nozzle and the variation of the flow with respect to the inflow conditions; when these vary the shock moves and the derivative of the flow variables with respect to inflow conditions is a Dirac measure and so the Taylor expansion does not exists. By extending the calculus of variation via the theory of distribution it is possible to show however that the derivatives exists. But the result has serious numerical implications, in particular it favors the mixed finite element methods. We shall give numerical illustrations using the finite element method for an inverse problem for a Darcy flow, for the design of a transonic nozzle and for the design of a business supersonic airplane for sonic boom minimization.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons