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We consider the nonlinear, inverse problem of identifying the stored energyfunction of a hyperelastic material from full knowledge of the displacementfield as well as from surface sensor measurements. The displacement field isrepresented as a solution of Cauchy's equation of motion, which is a nonlinear,elastic wave equation. Hyperelasticity means that the first Piola-Kirchhoffstress tensor is given as the gradient of the stored energy function. We assumethat a dictionary of suitable functions is available and the aim is to recoverthe stored energy with respect to this dictionary. The considered inverseproblem is of vital interest for the development of structural healthmonitoring systems which are constructed to detect defects in elastic materialsfrom boundary measurements of the displacement field, since the stored energyencodes the mechanical peroperties of the underlying structure. In this articlewe develope a numerical solver for both settings using the attenuated Landwebermethod. We show that the parameter-to-solution map satisfies the localtangential cone condition. This result can be used to prove local convergenceof the attenuated Landweber method in case that the full displacement field ismeasured. In our numerical experiments we demonstrate how to construct anappropriate dictionary and show that our algorithm is well suited to localizedamages in various situations.

Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method