Goal‐oriented adaptivity for parameter identification in linear micromorphic elasticity

Abstract

AbstractIt is well‐known that the classical Cauchy‐Boltzmann continuum, in spite of its simplicity, has its limitations, e.g. on the simulation of strain localization phenomena or size effects. For a remedy, one may resort to the so‐called generalized continuum theories, see e.g. [2] and references therein. Amongst others, we consider a class of higher order continua, namely the micromorphic continua, where the kinematics are enriched by means of a microstructure undergoing an affine micro deformation. As a drawback, the enriched kinematics lead to additional constitutive equations, increasing the number of material parameters, and therefore making the parameter identification procedure more involved. Here, the parameter identification is viewed as an inverse problem minimizing a least‐square function, see also [3] for a micropolar continuum. For simplicity, we restrict ourselves to a small strain theory. The increasing number of the degrees of freedom and the material parameters clearly motivates an application of adaptive methods. For an application on the direct micromorphic problems, we refer to [5]. Here for the inverse micromorphic problems, we follow the general ansatz by [1] for goal‐oriented error estimation and adaptivity, with which we control the spatial discretization errors of the finite element method. For illustration, numerical examples are presented.