Numerical analysis of a relaxed variational model of hysteresis in two-phase solids

This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with the finite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress field within one time-step. A posteriori error estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis

Saints and Orphans

Postcard from Brooke Carstensen, during the Linfield College January Term Program in Indi

Time- or State-Dependence? An Analysis of Inflation Dynamics using German Business Survey Data

This paper evaluates the predictions of different price setting theories using a new dataset constructed from a large panel of business surveys of German retail firms over the period 1970-2010. The dataset contains firm-specific information on both price realizations and expectations. Aggregating the price data we find clear evidence in favor of state-dependence; for periods of relatively high and volatile inflation not only the size of price changes (intensive margin) but also the fraction of price adjustment (extensive margin) is important for aggregate inflation dynamics. Moreover, at the business cycle frequency, variations in the extensive margin explain a large fraction of inflation variability even for moderate inflation periods. This holds both for price realizations and expectations suggesting a role for state-dependent sticky plan models. Moreover, results from a structural sign-restriction VAR model show that the extensive margin reacts significantly to a monetary policy shock and is more important for the response of overall inflation than the intensive margin conditional on the shock. These findings confirm the validity of state-dependent pricing models that stress the importance of the extensive margin - even for low inflation periods

Geometry of polycrystals and microstructure

We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-to-tetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations), but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.Comment: ESOMAT 2015 Proceedings, to appea

Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\￣Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory