Three-field mixed finite element methods for nonlinear elasticity

Neunteufel, Michael; Pechstein, Astrid S.; Schöberl, Joachim

doi:10.1016/j.cma.2021.113857

Cited by 9 publications

(10 citation statements)

References 42 publications

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“…In [22,33], a hybridization technique is proposed and analyzed for the geometrically linear and nonlinear case, respectively. It has been shown that the proposed hybridization improves the condition of the assembled stiffness matrix, and (for linear elastic problems after static condensation of the stresses) leads to a symmetric positive definitive system matrix.…”

Section: The Tdnns Methods For Linear Elastic Solidsmentioning

confidence: 99%

“…The Lagrangian multiplier λ is a vector valued finite element function defined uniquely on the element (inter-)faces pointing into the normal direction (λ × n = 0). The according finite element space can be implemented by using a facet space equipped with a normal vector [4,21,22]. The hybridized variational problem reads: find u, T, and λ, with u and λ satisfying the essential boundary conditions u t = 0 and λ n = 0 on Γ u such that…”

Section: The Tdnns Methods For Linear Elastic Solidsmentioning

confidence: 99%

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Mixed finite elements applied to acoustic wave problems in compressible viscous fluids under piezoelectric actuation

2022

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In the present contribution, we develop a mixed finite element method capable of the coupled multi-field simulation of a viscous fluid actuated by a piezoelectric resonator. Several challenges are met with in this setting, among which are the necessity of correct interface coupling, near incompressibility of the fluid, adverse geometric dimensions of flat piezoelectric transducers and different length scales of shear and pressure wave. Assuming small deformations and velocities, we present a mixed variational formulation with consistent interface coupling conditions in (mechanic) frequency domain. Both fluid and piezoelectric solid domain are discretized using Tangential-Displacement Normal-Normal-Stress elements. These elements model not only the deformation, but add an independent tensor-valued stress approximation. The method has been rigorously proven to be free from shear locking for flat prismatic or hexahedral elements. Thus, modeling of the flat geometry of piezoelectric resonators as well as resolution of the fastly decaying shear wave are facilitated. To circumvent the problem of volume locking due to the near incompressibility of the fluid, an additional independent pressure field is introduced. We present computational results indicating the capability of the method.

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Section: The Tdnns Methods For Linear Elastic Solidsmentioning

confidence: 99%

Section: The Tdnns Methods For Linear Elastic Solidsmentioning

confidence: 99%

Mixed finite elements applied to acoustic wave problems in compressible viscous fluids under piezoelectric actuation

2022

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“…Remark 3.1 (Relation with the TDNNS method). We specifically mention that the use of an H(curl)-conforming space V k,s h for the displacement field is inspired by the recent TDNNS method [17], where good performance for nonlinear elastostatics was obersved in [14] for several numerical examples including nearly incompressible materials and thick/thin structures. A major difference between the TDNNS method [14] and the current HDG method (23) is on the choice of the finite element spaces for the tensor fields P h and F h .…”

Section: 3mentioning

confidence: 99%

“…We specifically mention that the use of an H(curl)-conforming space V k,s h for the displacement field is inspired by the recent TDNNS method [17], where good performance for nonlinear elastostatics was obersved in [14] for several numerical examples including nearly incompressible materials and thick/thin structures. A major difference between the TDNNS method [14] and the current HDG method (23) is on the choice of the finite element spaces for the tensor fields P h and F h . We use a simple tensor space with standard pull-back mappings from the reference element for both tensor fields, and use the (positive) HDG stabilization term in the numerical flux to enforce the stability of the scheme.…”

Section: 3mentioning

confidence: 99%

“…We use a simple tensor space with standard pull-back mappings from the reference element for both tensor fields, and use the (positive) HDG stabilization term in the numerical flux to enforce the stability of the scheme. On the other hand, the TDNNS method [14] use a doublly Piola mapping for the symmetric part of the stress P h , and a doublly covariant mapping for the symmetric part of the deformation gradient F h where the stabilization parameter can be set to be zero; see more details in [14].…”

Section: 3mentioning

confidence: 99%

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Arbitrary Lagrangian–Eulerian hybridizable discontinuous Galerkin methods for incompressible flow with moving boundaries and interfaces

2020

Computer Methods in Applied Mechanics and Engineering

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We present a novel (high-order) hybridizable discontinuous Galerkin (HDG) scheme for the fluid-structure interaction (FSI) problem. The (moving domain) incompressible Navier-Stokes equations are discretized using a divergence-free HDG scheme within the arbitrary Lagrangian-Euler (ALE) framework. The nonlinear elasticity equations are discretized using a novel HDG scheme with an H(curl)-conforming velocity/displacement approximation. We further use a combination of the Nitsche's method (for the tangential component) and the mortar method (for the normal component) to enforce the interface conditions on the fluid/structure interface. A second-order backward difference formula (BDF2) is use for the temporal discretization. Numerical results on the classical benchmark problem by Turek and Hron [27] show a good performance of our proposed method.

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A mixed finite element formulation for elastoplasticity

Nagler

Pechstein

Humer

2022

Numerical Meth Engineering

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The present article, provides a novel mixed finite element formulation for elastoplasticity which is suitable for the discretization of thin-walled structures using highly anisotropic volume elements. We present a thermodynamically consistent framework for the modeling of elastoplastic stress response, where dissipative effects are considered through a dissipation function instead of explicit flow rules. Along these lines, a mixed incremental principle, in which stresses are included as independent unknowns, is derived. We propose to employ tangential-displacement normal-normal-stress (TDNNS) elements for a Galerkin discretization of the underlying variational problem. These elements were originally developed for linear elasticity, where it could be shown that they do not suffer from shear locking if the element's aspect ratio deteriorates. Excellent computational performance and accuracy of the proposed method are demonstrated in several benchmark problems.

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Three-field mixed finite element methods for nonlinear elasticity

Cited by 9 publications

References 42 publications

Mixed finite elements applied to acoustic wave problems in compressible viscous fluids under piezoelectric actuation

Mixed finite elements applied to acoustic wave problems in compressible viscous fluids under piezoelectric actuation

Arbitrary Lagrangian–Eulerian hybridizable discontinuous Galerkin methods for incompressible flow with moving boundaries and interfaces

A mixed finite element formulation for elastoplasticity

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