Multifield variational formulations of diffusion initial boundary value problems
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O R I G I NA L A RT I C L E
Jorge de Anda Salazar · Thomas Heuzé
· Laurent Stainier
Multifield variational formulations of diffusion initial boundary value problems
Received: 20 January 2020 / Accepted: 30 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We present two multifield and one single-field variational principles for the initial boundary value problem of diffusion. Chemical potential and concentration appear as conjugate variables in the multifield formulations. The main importance of the proposed formulations is the approach used to generate the variational principles, where the framework of Generalized Standard Materials is used for constitutive laws while natural boundary conditions and the balance of mass are used as constraints of the optimization problem. This approach allows to derive such principles for multiphysic problems in a generic manner. A detailed derivation and analysis of the formulations are presented, where it can be seen their equivalence with the most common strong and weak forms of the problem using Fick’s laws along with the logarithmic mass action law. From the stationarity condition with respect to the mass flux of the initially proposed functional, two main relations are identified. First, the chemical potential appears as the opposite of the Lagrange multiplier that allows to enforce the balance of mass and natural boundary conditions. Second, a conjugate relation is found for a given substance between its mass flux and the opposite of the gradient of its chemical potential. Furthermore, to reduce the number of variables of the initial variational principle, a field reduction is applied, reaching the model presented by Miehe et al. (Int J Numer Methods Eng 99(10):737, 2014. https://doi.org/10.1002/nme.4700) for Fickean diffusion. Nevertheless, the aforementioned relations cannot be derived from the reduced model. Finally, a numerical implementation is presented for completeness where we compare the performance of the proposed formulations against the usual weak form. Keywords Multifield diffusion · Variational formulation · Fick’s laws · Logarithmic mass action law · Coupled problem
1 Introduction In his seminal paper [1], Adolf Fick established the mathematical structure of the diffusion process, giving rise to what is called Fick’s laws. This set of equations, which uses the concentration c and the mass flux j Communicated by Andreas Oechsner. J. de Anda Salazar · T. Heuzé (B) · L. Stainier Institut de Recherche en Génie Civil et Mécanique (GeM, UMR 6183 CNRS/ECN/UN), École Centrale Nantes, 1 rue de la Noë, BP 92101, 44321 Nantes, France E-mail: [email protected] J. de Anda Salazar E-mail: [email protected] L. Stainier E-mail: [email protected]
J. de Anda Salazar et al.
as state variables, constitutes the most general way to model diffusion. Nevertheless, a wide variety of models can be derived to describe more specialized diffusion processes. Some of these more specialized processes of diffusio
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