This paper presents a stabilized mixed finite element method for advection-diffusion-reaction phenomena that involve an anisotropic viscous fluid diffusing and chemically reacting with an anisotropic elastic solid. The reactive fluid–solid mixture theory of Hall and Rajagopal (Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory. Math Mech Solid 2012; 17: 131–164) is employed wherein energy and entropy production relations are captured via an equation describing the Lagrange multiplier that results from imposing the constraint of maximum rate of entropy production. The primary partial differential equations are thus reduced to the balance of mass and balance of linear momentum equations for the fluid and the solid, together with an equation for the Lagrange multiplier. Present implementation considers a simplification of the full system of governing equations in the context of isothermal problems, although anisothermal studies are being investigated. The method is applied to problems involving Fickian diffusion, oxidation of PMR-15 polyimide resin, and slurry infiltration, within a one-dimensional finite element context. Results of the oxidation modeling of Tandon et al. (Modeling of oxidative development in PMR-15 resin. Polym Degrad Stab 2006; 91: 1861–1869) are recovered by employing the reaction kinetics model and properties assumed there; the only additional assumed properties are two constants describing coupled chemomechanical and purely chemical dissipation, and standard values for viscosity of air and PMR-15 stiffness properties. The present model provides the individual constituent kinematic and kinetic behaviors, thus adding rich detail to the interpretation of the process in comparison to the original treatment. The last problem considered is slurry infiltration that demonstrates the applicability of the model to account for the imposed mass deposition process and consequent effects on the kinematic and kinetic behaviors of the constituents.

This paper presents B-splines and nonuniform rational B-splines (NURBS)-based finite element method for self-consistent solution of the Schrödinger wave equation (SWE). The new equilibrium position of the atoms is determined as a function of evolving stretching of the underlying primitive lattice vectors and it gets reflected via the evolving effective potential that is employed in the SWE. The nonlinear SWE is solved in a self-consistent fashion (SCF) wherein a Poisson problem that models the Hartree and local potentials is solved as a function of the electron charge density. The complex-valued generalized eigenvalue problem arising from SWE yields evolving band gaps that result in changing electronic properties of the semiconductor materials. The method is applied to indium, silicon, and germanium that are commonly used semiconductor materials. It is then applied to the material system comprised of silicon layer on silicon–germanium buffer to show the range of application of the method.

Summary A mixture‐theory‐based model for multi‐constituent solids is presented where each constituent is governed by its own balance laws and constitutive equations. Interactive forces between constituents that emanate from maximization of entropy production inequality provide the coupling between constituent‐specific balance laws and constitutive models. The deformation of multi‐constituent mixtures at the Neumann boundaries requires imposing inter‐constituent coupling constraints such that the constituents deform in a self‐consistent fashion. A set of boundary conditions is presented that accounts for the non‐zero applied tractions, and a variationally consistent method is developed to enforce inter‐constituent constraints at Neumann boundaries in the finite deformation context. The new method finds roots in a local multiscale decomposition of the deformation map at the Neumann boundary. Locally satisfying the Lagrange multiplier field and subsequent modeling of the fine scales via edge bubble functions result in closed‐form expressions for a generalized penalty tensor and a weighted numerical flux that are free from tunable parameters. The key novelty is that the consistently derived constituent coupling parameters evolve with material and geometric nonlinearity, thereby resulting in optimal enforcement of inter‐constituent constraints. Various benchmark problems are presented to validate the method and show its range of application. Copyright © 2016 John Wiley & Sons, Ltd.

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