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Sensitivity and approximation of coupled fluid-structure equations by virtual control method. (English) Zbl 1136.74319

Summary: 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.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
49K40 Sensitivity, stability, well-posedness
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74M05 Control, switches and devices (“smart materials”) in solid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76M30 Variational methods applied to problems in fluid mechanics

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