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Mixed frameworks and structure preserving integration for coupled electro‐elastodynamics
Author(s) -
Franke Marlon,
Ortigosa Rogelio,
Janz Alexander,
Gil Antonio J.,
Betsch Peter
Publication year - 2019
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201900014
Subject(s) - integrator , mathematics , kinematics , finite element method , nonlinear system , tensor (intrinsic definition) , mathematical analysis , classical mechanics , geometry , physics , voltage , quantum mechanics , thermodynamics
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 thermo‐elastodynamics, 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.