A mixed variational framework for the design of energy–momentum schemes inspired by the structure of polyconvex stored energy functions

Betsch, Peter; Janz, Alexander; Hesch, Christian

doi:10.1016/j.cma.2018.01.013

Cited by 30 publications

(80 citation statements)

References 15 publications

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“…[1]) where ε ijk denotes the permutation symbol and the summation convention applies. Furthermore the tensor cross product is defined by (A B) ij = ε iαβ ε jab A αa B βb (for more informations see e.g.…”

Section: Polyconvexity Inspired Frameworkmentioning

confidence: 99%

“…A beneficial cascade form of the constraints (see [1]) A beneficial cascade form of the constraints (see [1])…”

Section: Electro-elastodynamicsmentioning

confidence: 99%

“…Polyconvexity inspired energy functionals are obtained by using the rediscovered tensor cross product which greatly simplifies the algebra, see [1]. Polyconvexity inspired energy functionals are obtained by using the rediscovered tensor cross product which greatly simplifies the algebra, see [1].…”

mentioning

confidence: 99%

“…Polyconvexity inspired energy functionals are obtained by using the rediscovered tensor cross product which greatly simplifies the algebra, see [1]. In particular in [1] an elegant cascade system of kinematic constraints was introduced for elastodynamics, crucial for the satisfaction of the required conservation properties of a new family of energy momentum (EM) consistent time integrators. On this basis coupled problems like e.g.…”

mentioning

confidence: 99%

“…In this connection an extended kinematic set, consisting of the right Cauchy-Green tensor, its co-factor and its Jacobian, is introduced. The objective of the present contribution is the introduction of new mixed variational principles for EM consistent time integrators in electro-elastodynamics, hence bridging the gap between the previous works [3] and [1], opening the possibility to a variety of new finite element implementations, see [4]. non-linear thermoelastodynamics, see [2] or electro-elastodynamics, see [3], can be considered.…”

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confidence: 99%

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Mixed frameworks and structure preserving integration for coupled electro‐elastodynamics

Franke

Ortigosa

Janz

et al. 2019

Proc Appl Math and Mech

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In the present contribution new approaches for the design of structure preserving time integrators for nonlinear coupled problems are proposed. Polyconvexity inspired energy functionals are obtained by using the rediscovered tensor cross product which greatly simplifies the algebra, see [1]. In this connection an extended kinematic set, consisting of the right Cauchy-Green tensor, its co-factor and its Jacobian, is introduced. On this basis coupled problems like e.g. non-linear thermoelastodynamics, see [2] or electro-elastodynamics, see [3], can be considered. Furthermore the formulations are readily extendible for mixed Hu-Washizu type formulations where the extended kinematic set is introduced as unknown field. In particular in [1] an elegant cascade system of kinematic constraints was introduced for elastodynamics, crucial for the satisfaction of the required conservation properties of a new family of energy momentum (EM) consistent time integrators. The objective of the present contribution is the introduction of new mixed variational principles for EM consistent time integrators in electro-elastodynamics, hence bridging the gap between the previous works [3] and [1], opening the possibility to a variety of new finite element implementations, see [4]. The following characteristics of the proposed EM consistent time integrator make it very appealing: (i) the new family of time integrators can be shown to be thermodynamically consistent and second order accurate; (ii) piecewise discontinuous interpolation of the mixed fields is carried out in order to obtain a computational cost comparable to that of standard displacement, electric potential formulations. Eventually, the superior numerical performance of the proposed formulation is demonstrated.

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Section: Polyconvexity Inspired Frameworkmentioning

confidence: 99%

“…A beneficial cascade form of the constraints (see [1]) A beneficial cascade form of the constraints (see [1])…”

Section: Electro-elastodynamicsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Mixed frameworks and structure preserving integration for coupled electro‐elastodynamics

Franke

Ortigosa

Janz

et al. 2019

Proc Appl Math and Mech

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An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics

Franke

Janz

Schiebl

et al. 2018

Numerical Meth Engineering

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The present contribution provides a new approach to the design of energy momentum consistent integration schemes in the field of nonlinear thermo-elastodynamics. The method is inspired by the structure of polyconvex energy density functions and benefits from a tensor cross product that greatly simplifies the algebra. Furthermore, a temperature-based weak form is used, which facilitates the design of a structure-preserving time-stepping scheme for coupled thermoelastic problems. This approach is motivated by the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework for open systems. In contrast to complex projection-based discrete derivatives, a new form of an algorithmic stress formula is proposed. The spatial discretization relies on finite element interpolations for the displacements and the temperature. The superior performance of the proposed formulation is shown within representative quasi-static and fully transient numerical examples. KEYWORDSfinite element method, nonlinear thermo-elastodynamics, polyconvexity-based framework, structure-preserving discretization, tensor cross product INTRODUCTIONThe present contribution aims for the consistent discretization of nonlinear thermo-elastodynamics. The emphasis of this paper is on both theoretical and numerical aspects. In recent decades, thermomechanical constitutive models have been addressed in numerous works (see, eg, the works of Miehe, 1 Holzapfel and Simo, 2 and Reese and Govindjee 3 as well as textbooks by Holzapfel 4 and Gonzalez and Stuart 5 ). Classically, the hyperelastic Helmholtz free-energy density function depends only on the deformation gradient and the temperature. Furthermore, the weak form is deduced from its strong form. Dependent on the chosen material model, eg, for a Mooney-Rivlin model, the consistent linearization may lead to cumbersome expressions. In contrast to the classic approach, the present work is inspired by the concept of polyconvexity (see, eg, the works of Ball 6 and Ciarlet 7 ), where the Helmholtz free energy is a convex function of the deformation gradient, its cofactor, and its determinant and is concave with respect to absolute temperature. In addition to the polyconvexity-based framework, the present work relies on the cross product between second-order tensors, as introduced in the work of de Boer 8 (see also Appendix B 4.9.3 in another work of de Boer 9 ). This tensor cross product has been used in the context of large-strain Int J Numer Methods Eng. 2018;115:549-577. wileyonlinelibrary.com/journal/nme 549 550 FRANKE ET AL.hyperelasticity in the works of Bonet et al 10,11 and remarkably simplifies the algebraic manipulation of the large-strain continuum formulation and is extended herein to the thermomechanical formulation. Furthermore, the polyconvexity-based framework makes possible a wide range of advanced finite element technologies in the context of continuum mechanics. For example, the introduction of mixed finite elements, 10,12 phase-field fracture models, 13 anisotropic...

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On transformations and shape functions for enhanced assumed strain elements

Pfefferkorn

Betsch

2019

Numerical Meth Engineering

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Summary We summarize several previously published geometrically nonlinear EAS elements and compare their behavior. Various transformations for the compatible and enhanced deformation gradient are examined. Their effect on the patch test is one main concern of the work, and it is shown numerically and with a novel analytic proof that the improved EAS element proposed by Simo et al in 1993 does not fulfill the patch test. We propose a modification to overcome that drawback without losing the favorable locking‐free behavior of that element. Furthermore, a new transformation for the enhanced field is proposed and motivated in a curvilinear coordinate frame. It is shown in numerical tests that this novel approach outperforms all previously introduced transformations.

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A mixed variational framework for the design of energy–momentum schemes inspired by the structure of polyconvex stored energy functions

Cited by 30 publications

References 15 publications

Mixed frameworks and structure preserving integration for coupled electro‐elastodynamics

Mixed frameworks and structure preserving integration for coupled electro‐elastodynamics

An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics

On transformations and shape functions for enhanced assumed strain elements

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