High-order mixed finite elements for an energy-based model of the polarization process in ferroelectric materials

An energy-based model of the ferroelectric polarization process is presented in the current contribution. In an energy-based setting, dielectric displacement and strain (or displacement) are the primary independent unknowns. As an internal variable, the remanent polarization vector is chosen. The model is then governed by two constitutive functions: the free energy function and the dissipation function. Choices for both functions are given. As the dissipation function for rate-independent response is non-differentiable, it is proposed to regularize the problem. Then, a variational equation can be posed, which is subsequently discretized using conforming finite elements for each quantity. We point out which kind of continuity is needed for each field (displacement, dielectric displacement and remanent polarization) is necessary to obtain a conforming method, and provide corresponding finite elements. The elements are chosen such that Gauss' law of zero charges is satisfied exactly. The discretized variational equations are solved for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve

Advanced mechanical simulation models for automatic panel benders

With automatic panel benders complete products are manufactured from sheet metal. In order to achieve short cycle times with high flexibility, a deep insight into the non-linear bending process is required. For this reason, efficient mechanical simulation models have been implemented, combining Finite Element Method, multibody dynamics simulation tools, contact mechanics algorithms and substructuring. Scope of this work is the comparison of several simulation models with measurement results performed on a Salvagnini P4XeD automatic panel bender

An analysis of the TDNNS method using natural norms

The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space H(curl) for the displacement. We define the space H(divdiv) for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the H(curl) and a discrete H(divdiv) norm. A-priori error estimates of optimal order with respect to these norms are obtained. Beside providing a new analysis for the elasticity equation, the numerical techniques developed in this paper are a foundation for more complex models from structural mechanics such as Reissner Mindlin plate equations, see Pechstein and Schöberl (Numerische Mathematik 137(3):713740, 2017).(VLID)341800

Exact solutions for the buckling and postbuckling of a shear-deformable cantilever subjected to a follower force

The buckling and postbuckling of a shear-deformable cantilever is studied using Reissners geometrically exact relations for the planar deformation of beams. The cantilever is subjected to a compressive follower force whose line of action passes through a spatially fixed point. To study the buckling behavior, a consistent linearization of equilibrium and kinematic relations is introduced. The influence of shear deformation and extensibility on the critical loads is studied. The buckling behavior turns out to crucially depend on the ratio between the shear stiffness and the extensional stiffness of the structure. Closed-form solutions in terms of elliptic integrals for buckled configurations of the cantilever are derived in the present paper.(VLID)440537