Inverse problems in the mechanical characterization of elastic arteries
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Introduction Inverse problems posed by the mechanical characterization of materials Identification of mechanical properties is crucial for all kinds of materials in order to develop faithful models of solids and structures, predict their mechanical response to a given loading, or assess their integrity and monitor their health. The mathematical problems posed by the identification of material properties are often referred to as inverse problems. To define an inverse problem, it is convenient to first define its opposite: a forward problem.1 In mechanics, solving a forward problem means predicting the result of a mechanical action on a solid (displacement, strain, and stress) from knowledge of the material model and boundary conditions, which are combined in a boundary-value problem of partial differential equations based on the local mechanical equilibrium. On the other hand, an inverse problem is posed when the result of the mechanical action is partly or fully measured and one wants to employ these measurements to determine unknown parameters of the material model, unknown elements of the boundary conditions, or the unknown initial geometry of the solid before the mechanical action.2 Inverse problems should not be confused with semi-inverse problems, which are a sub-category of forward problems. Semi-inverse problems have an exact analytical solution,
whereas the majority of forward problems have only approximate solutions that can be computed numerically using (e.g., the finite-element method). Semi-inverse problems occur especially when predicting the result of a mechanical action on solids with simple geometries.3 When the result of the mechanical action on such solids is measured and one wants to employ these measurements to determine unknown parameters of the material model, the closed-form expressions of the mechanical fields allow a simpler identification of the unknown material parameters. This subcategory of inverse problems may be classed as semi-forward problems. Semi-forward problems occur in a number of traditional mechanical tests, often called statically determined tests, where the parameters can be estimated by best-fit determination from the data. Solving inverse problems implies the definition of a cost function, estimating the distance between the model predictions and measurements. The cost function is minimized using either a least-squares technique (such as the LevenbergMarquardt algorithm) or a genetic algorithm, except in the case of semi-forward problems and linear least-squares which exhibit an explicit solution. In general situations, the model is solved numerically using a finite-element model updating technique (FEMU). In specific situations, when full-field measurements are available, an alternative to FEMU is possible in
Claire Morin, École Nationale Supérieure des Mines, Saint-Etienne, France; [email protected] Stéphane Avril, École Nationale Supérieure des Mines, Saint-Etienne, France; [email protected] DOI: 10.1557/mrs.2015.63
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