2015/11/13 by Julia Seydel, Thomas Schuster, Seydel, Julia +1 · 1 citation
Earth and Planetary Sciences · Engineering · Mathematics · #35L70 #65M32 #74B20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Seismic Imaging and Inversion Techniques #Ultrasonics and Acoustic Wave Propagation
paper · pdf · doi:10.48550/arxiv.1511.04259
openalex publication_date 2015/11/13 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We consider the nonlinear,inverse problem of computing the stored energy\nfunction of a hyperelastic material from the full knowledge of the displacement\nfield. The displacement field is described as solution of the nonlinear,\ndynamic, elastic wave equation, where the first Piola-Kirchhoff stress tensor\nis given as the gradient of the stored energy function. We assume that we have\na dictionary at hand such that the energy function is given as a conic\ncombination of the dictionary's elements. In that sense the mathematical model\nof the direct problem is the nonlinear operator that maps the vector of\nexpansion coefficients to the solution of the hyperelastic wave equation. In\nthis article we summarize some continuity results for this operator and deduce\nits Fr 'echet derivative as well as the adjoint of this derivative. Since the\nstored energy function encodes mechanical properties of the underlying,\nhyperelastic material, the considered inverse problem is of highest interest\nfor structural health monitoring systems where defects are detected from\nboundary measurements of the displacement field. For solving the inverse\nproblem iteratively by the Landweber method or Newton type methods, the\nknowledge of the Fr 'echet derivative and its adjoint is of utmost\nimportance.\n