Appl Math Optim 52:183–218 (2005)
2005 Springer Science+Business Media, Inc.
Sensitivity and Approximation of Coupled Fluid–Structure
Equations by Virtual Control Method
Cornel Marius Murea
and Carlos V´azquez
Laboratoire de Math´ematiques, Universit´e de Haute-Alsace,
4 rue des Fr`eres Lumi`ere, 68093 Mulhouse, France
Departamento de Matem´aticas, Facultad de Inform´atica,
Universidade da Coru˜na,
Campus Elvi˜na s/n, 15071 La Coru˜na, Spain
Abstract. The formulation of a particular ﬂuid–structure interaction as an optimal
control problem is the departure point of this work. The control is the vertical
component of the force acting on the interface and the observation is the vertical
component of the velocity of the ﬂuid on the interface. This approach permits
us to solve the coupled ﬂuid–structure problem by partitioned procedures. The
analytic expression for the gradient of the cost function is obtained in order to devise
accurate numerical methods for the minimization problem. Numerical results arising
from blood ﬂow in arteries are presented. To solve the optimal control problem
numerically, we use a quasi-Newton method which employs the analytic gradient
of the cost function and the approximation of the inverse Hessian is updated by the
Broyden, Fletcher, Goldforb, Shano (BFGS) scheme. This algorithm is faster than
ﬁxed point with relaxation or block Newton methods.
Key Words. Fluid–structure interaction, Virtual control, Mixed formulations,
Optimization algorithms, Sensitivity analysis.
AMS Classiﬁcation. 74F10, 49K40, 65N30.
This work has been partially supported by the Research Projects of Xunta de Galicia (PGIDIT-02-
PXIC-10503-PN), D.G.E.S. (BFM2001-3261-C02-02) and M.E.C. (MTM2004-05796-C02-01). Moreover,
C. M. Murea has been granted a post of “Profesor Visitante” at the University of La Coruna by Xunta de