16 research outputs found
Numerische Mathematik / An analysis of the TDNNS method using natural norms
No full textThe 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):713–740, 2017).Refereed/Peer-reviewedVersion of recor
Three-field mixed finite element methods for nonlinear elasticity
No full textIn this paper, we extend the tangential-displacement normal-normal-stress continuous (TDNNS) method from Pechstein and Schöberl (2011) to nonlinear elasticity. By means of the Hu-Washizu principle, the distributional derivatives of the displacement vector are lifted to a regular strain tensor. We introduce three different methods, where either the deformation gradient, the Cauchy-Green strain tensor, or both of them are used as independent variables. Within the linear sub-problems, all stress and strain variables can be locally eliminated leading to an equation system in displacement variables, only. The good performance and accuracy of the presented methods are demonstrated by means of several numerical examples
The TDNNS method for Reissner–Mindlin plates
No full textA new family of locking-free finite elements for shear deformable Reissner–Mindlin plates is presented. The elements are based on the “tangential-displacement normal-normal-stress” formulation of elasticity. In this formulation, the bending moments are treated as separate unknowns. The degrees of freedom for the plate element are the nodal values of the deflection, tangential components of the rotations and normal–normal components of the bending strain. Contrary to other plate bending elements, no special treatment for the shear term such as reduced integration is necessary. The elements attain an optimal order of convergence.Refereed/Peer-reviewedVersion of recor
Acta Mechanica / Exact solutions for the buckling and postbuckling of a shear-deformable cantilever subjected to a follower force
No full textThe buckling and postbuckling of a shear-deformable cantilever is studied using Reissner’s 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.Refereed/Peer-reviewedVersion of recor
The polarization process of ferroelectric materials in the framework of variational inequalities
No full textWe are concerned with the mathematical modeling of the polarization process in ferroelectric media. We assume that this dissipative process is governed by two constitutive functions, which are the free energy function and the dissipation function. The dissipation function, which is closely connected to the dissipated energy, is usually non‐differentiable. Thus, a minimization condition for the overall energy includes the subdifferential of the dissipation function. This condition can also be formulated by way of a variational inequality in the unknown fields strain, dielectric displacement, remanent polarization and remanent strain. We analyze the mathematical well‐posedness of this problem. We provide an existence and uniqueness result for the time‐discrete update equation. Under stronger assumptions, we can prove existence of a solution to the time‐dependent variational inequality. To solve the discretized variational inequality, we use mixed finite elements, where mechanical displacement and dielectric displacement are unknowns, as well as polarization (and, if included in the model, remanent strain). It is then possible to satisfy Gauss' law of zero free charges exactly. We propose to regularize the dissipation function and solve for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve.Linz Center of Mechatronics COMET-K2Version of recor
An Equilibration Based A Posteriori Error Estimate for the Biharmonic Equation and Two Finite Element Methods
No full textWe develop an a posteriori error bound for the interior penalty discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor is that of symmetric tensor fields with continuous normal-normal components, and is well-known from the Hellan-Herrmann-Johnson mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original Hellan-Herrmann-Johnson formulation, which directly provides an equilibrated moment tensor
Continuum Modeling and Finite Element Simulation of Incompressible Dielectric Viscoelastic Actuators at Finite Strains
No full textDielectric elastomers, known for their ability to undergo large deformations exceeding 100%, are widely used as actuators in adaptive structures and soft robotics. Within the current contribution, we present a continuum material model that captures the incompressibility and viscous behavior of these polymers under finite s train and e lectric a ctuation. To address l arge deformations, we use a multiplicative decomposition of the deformation gradient to separate elastic and viscous effects. The elastic response is represented by a Yeoh potential, which is well suited to describe the material behavior under large strains. The evolution of internal strains is modeled using a dissipation function. Electric field a nd d ielectric d isplacement a re modeled i n s patial c onfiguration, le ading to an electromechanically coupled problem. We propose a mixed finite e lement f ormulation w ithin a variational framework based on the above thermodynamic principles. We introduce a novel approach using volume-preserving tensor-valued elements for internal strains, where we make use of matrix exponential functions to achieve incompressiblity exactly. As an example, we consider an experimental setup of a three-dimensional circular actuator. We provide material parameters for VHB4910 for the proposed model, and compare our results to experimental data from a different work
High-order mixed finite elements for an energy-based model of the polarization process in ferroelectric materials
No full textAn 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
A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation
No full textActa Mechanica / Nonlinear electromechanical coupling in ferroelectric materials : large deformation and hysteresis
No full textSmart materials respond to external stimuli, e.g., electric fields, which enables their use as sensors and actuators. The electromechanical coupling of the direct and converse piezoelectric effects, for instance, is used for both actuation and sensing in diverse engineering applications. The response of ferroelectric materials depends on their state of remanent polarization and the presence of an external electric field. To extend the operational range of sensors and actuators, an accurate understanding of the evolution of the material’s state of polarization is imperative, which requires both physical and geometric nonlinearities to be taken into account. Moreover, polymeric smart materials like PVDF allow significantly larger deformation as compared to conventional piezoelectric ceramics. The electromechanical coupling in piezoelectric materials manifests in ferroelectric and ferroelastic hystereses, which are related to both reversible and irreversible processes. Focusing on the latter, we transfer phenomenological models for domain switching in ferroelectric materials to the geometrically nonlinear regime. For this purpose, we follow related concepts of geometrically nonlinear elastoplasticity, where the concept of a multiplicative decomposition of the deformation gradient plays a key role. Accordingly, an additional deformation path that describes the evolution of the poled state from the unpoled referential configuration is introduced. The constitutive response of the material to mechanical and electrical loads is discussed, and dissipative internal forces that drive the evolution of the remanent polarization are derived within a thermodynamical framework and the principle of maximum dissipation.Refereed/Peer-reviewedVersion of recor
