3D mixed finite elements for curved, flat piezoelectric structures

Abstract

The Tangential-Displacement Normal-Normal-Stress (TDNNS) method is a finite element method that was originally introduced for elastic solids and later extended to piezoelectric materials. It uses tangential components of the displacement and normal components of the normal stress vector as degrees of freedom for elasticity. For the electric field, the electric potential is used. The TDNNS method has been shown to provide elements which do not suffer from shear locking. Therefore thin structures (e.g. piezoelectric patch actuators) can be modeled efficiently. Hexahedral and prismatic elements of arbitrary polynomial order are provided in the current contribution. We show that these elements can be used to discretize curved, shell-like geometries by curved elements of high aspect ratio. The order of geometry approximation can be chosen independently from the polynomial order of the shape functions. We present two examples of curved geometries, a circular patch actor and a radially polarized piezoelectric semi-cylinder. Simulation results of the TDNNS method are compared to results gained in ABAQUS. We obtain good results for displacements and electric potential as well as for stresses, strains and electric field when using only one element in thickness direction.

Variational inequalities for ferroelectric constitutive modeling

“…To do so, a mixed finite element scheme is used. The specific approach taken in this work uses tangential displacements and normal components of the normal stress vector ( σ · n ) · n as degrees of freedom, which motivate the abbreviation “TDNNS.” This method was originally developed for elastic solids by Pechstein and Schöberl (2011, 2012) and later extended for linear piezoelectric materials (Meindlhumer and Pechstein, 2018; Pechstein et al, 2018) and for and for geometrically nonlinear electro-mechanically coupled problems (Pechstein, 2019). As the TDNNS method does not suffer from locking effects for elements of arbitrary aspect ratios, it is highly suitable for the discretization of thin structures.…”

“…This work is based on Pechstein et al (2018) and Meindlhumer and Pechstein (2018) for linear piezoelectric materials. Note that, for the mixed TDNNS method, the d -tensor formulation can be used directly, and there is no need to transfer the electric permittivity at constant stress ϵ σ to that at constant strain ϵ ε algebraically, as is necessary in standard formulations.…”

“…We use non-standard TDNNS elements, where tangential displacements and the normal component of the normal stress are chosen as degrees of freedom (Pechstein and Schöberl, 2011, 2018). These elements were adapted to linear piezoelectricity in Pechstein et al (2018) and Meindlhumer and Pechstein (2018). The TDNNS elements have been shown to be free from locking effects and therefore they are highly suitable for the efficient discretization of thin structures, which will be shown in one particular example.…”

Fluid-Structure Interaction Response of a Water Conveyance System with a Surge Chamber during Water Hammer

“…In this study, f = F(Re), Re is Reynolds number, When Re < 2100, f = 16/Re, when 3000 < Re < 10000, f = 0.079/Re 0. 25 When Re > 105, f = 0.046/Re 0.2 [25,26] Here, Re = ρ f |V|d/ν, v is Kinematic viscosity of liquid.…”

Three-field mixed finite element methods for nonlinear elasticity

“…The stress space Σ consists of the normal-normal continuous elements introduced in [25]. For two-dimensional domains, triangular and quadrilateral elements have been introduced, while for three-dimensional meshes tetrahedra, hexahedra and prismatric elements have been developed so far, see [25,26,22]. The so-called Regge finite element space is constructed such that the tangential-tangential part C T T := (I − N ⊗ N )C(I − N ⊗ N ), one component in 2D and four in 3D, is continuous.…”