Identification of unknown spatial load distributions in a vibrating Euler–Bernoulli beam from limited measured data. (4th April 2016) - Hasanov, Alemdar; Kawano, Alexandre
Abstract: Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation ρ ( x ) w tt + μ ( x ) w t + ( EI ( x ) w xx ) xx − T r u xx = ∑ m = 1 M g m ( t ) f m ( x ), ( x, t ) ∊ Ω T ≔ ( 0, l ) × ( 0, T ), with hinged–clamped ends ( w ( 0, t ) = w xx ( 0, t ) = 0, w ( l, t ) = w x ( l, t ) = 0, t ∊ ( 0, T ) ), are studied. Here g m ( t ) are linearly independent functions, describing an asynchronous temporal loading, and f m ( x ) are the spatial load distributions. In the first identification problem the values ν k ( t ), k = 1, K ̄, of the deflection w ( x, t ), are assumed to be known, as measured output data, in a neighbourhood of the finite set of points P ≔ x k ∊ ( 0, l ), k = 1, K ̄ ⊂ ( 0, l ), corresponding to the internal points of a continuous beam, for all t ∊ ] 0, T [ . In the second identification problem the values θ k ( t ), k = 1, K ̄, of the slope w x ( x, t ), are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all t ∊ ] 0, T [ . These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data ν k ( t ) and θ k ( t ) contain random noise. In the first part of the study we prove that each measured output data ν k ( t ) and θ k ( t ), k = 1, K ̄ can uniquely determine the unknown functions f m ∊ H − 1 ( ] 0, l [ ), m = 1, M ̄ . In the second part of the study we will introduce the input–output operators  d : L 2 ( 0, T ) ↦ L 2 ( 0, T ), (  d f ) ( t ) ≔ w ( x, t ; f ), x ∊ P, f ( x ) ≔ ( f 1 ( x ), …, f M ( x ) ), and  s : L 2 ( 0, T ) ↦ L 2 ( 0, T ), (  s f ) ( t ) ≔ w x ( x, t ; f ), x ∊ P, corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations:  d f = ν and  s f = θ, where ν ( t ) ≔ ( ν 1 ( t ), …, ν K ( t ) ) and θ k ( t ) ≔ ( θ 1 ( t ), …, θ K ( t ) ) . Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals J 1 α ( f ) ≔ ( 1 / 2 ) ∥  d f − ν ∥ L 2 ( 0, T ) 2 + ( 1 / 2 ) α ∥ f ∥ L 2 ( 0, T ) 2 and J 2 α ( f ) ≔ ( 1 / 2 ) ∥  s f − θ ∥ L 2 ( 0, T ) 2 + ( 1 / 2 ) α ∥ f ∥ L 2 ( 0, T ) 2 . Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Fréchet gradients J 1 ′ ( f ) and J 2 ′ ( f ) of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads g m ( t ) . Moreover, we show that these gradients are Lipschitz continuous, which allows the use of gradient type iteration convergent algorithms. Two applications of the proposed theory are presented. It is shown that solvability results for inverse source problems related to the synchronous loading case, with a single interior measured data, are special cases of the obtained results for asynchronously distributed spatial load cases.