Inverse problems in elasticity

This review is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters or buried objects such as cracks. These inverse problems are considered mainly for three-dimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e., fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded

Residual equidistribution of modular symbols and cohomology classes for quotients of hyperbolic nn-space

We provide a new and simple automorphic method using Eisenstein series tostudy the equidistribution of modular symbols modulo primes, which we apply toprove an average version of a conjecture of Mazur and Rubin. More precisely, weprove that modular symbols corresponding to a Hecke basis of weight 2 cuspforms are asymptotically jointly equidistributed mod $p$ while we allowrestrictions on the location of the cusps. As an application, we obtain aresidual equidistribution result for Dedekind sums. Furthermore, we calculatethe variance of the distribution and show a surprising bias with connections toperturbation theory. Additionally, we prove the full conjecture in someparticular cases using a connection to Eisenstein congruences. Finally, ourmethods generalise to equidistribution results for cohomology classes of finitevolume quotients of $n$-dimensional hyperbolic space.<br