Abstract
This paper is devoted to the mathematical and numerical analysis of a mixed-mixed PDE system describing the stress-assisted diffusion of a solute into an elastic material. The equations of elastostatics are written in mixed form using stress, rotation and displacements, whereas the diffusion equation is also set in a mixed three-field form, solving for the solute concentration, for its gradient, and for the diffusive flux. This setting simplifies the treatment of the nonlinearity in the stress-assisted diffusion term. The analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of Schauder and Banach fixed-point theorems together with the Babuška–Brezzi and Lax–Milgram theories. Concerning numerical discretization, we propose two families of finite element methods, based on either PEERS or Arnold–Falk–Winther elements for elasticity, and a Raviart–Thomas and piecewise polynomial triplet approximating the mixed diffusion equation. We prove the well-posedness of the discrete problems, and derive optimal error bounds using a Strang inequality. We further confirm the accuracy and performance of our methods through computational tests.
Highlights
We are interested in the mathematical and numerical study of a stationary problem representing diffusion–deformation processes where the stress acts as a coupling variable
When lithium diffuses into a secondary particle, its core expands and its elastic response, with that of neighbouring particles and the surrounding electrolyte, modify the diffusive properties inside the medium
Under similar assumptions we deduce error bounds in Section 5; and we close in Section 6 with a numerical example that confirms the theoretical rates of convergence, and a second test studying the applicability of the discrete formulation in the simulation of 3D microscopic lithiation processes
Summary
We are interested in the mathematical and numerical study of a stationary problem representing diffusion–deformation processes where the stress acts as a coupling variable. Ruiz-Baier / Computers and Mathematics with Applications 77 (2019) 1312–1330 of binder in electrodes [14], and a general local–global well-posedness theory [15]. From these approaches, in [16] we have recently proposed a mixed-primal formulation for stress-assisted diffusion. The model covers the linear elastic regime, it incorporates the rotation tensor as supplementary variable serving to impose stress symmetry in a weak manner; and this mixed problem is coupled with a primal formulation for diffusion. Under similar assumptions we deduce error bounds in Section 5; and we close in Section 6 with a numerical example that confirms the theoretical rates of convergence, and a second test studying the applicability of the discrete formulation in the simulation of 3D microscopic lithiation processes
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