Multi-scale homogenization of moving interface problems with flux jumps: application to solidification

Abstract

In this paper, a multi-scale analysis scheme for solidification based on two-scale computational homogenization is discussed. Solidification problems involve evolution of surfaces coupled with flux jump boundary conditions across interfaces. We provide consistent macro-micro transition and averaging rules based on Hill's macrohomogeneity condition. The overall macro-scale behavior is analyzed with solidification at the micro-scale modeled using an enthalpy formulation. The method is versatile in the sense that two different models can be employed at the macroand micro-scales. The micro-scale model can incorporate all the physics associated with solidification including moving interfaces and flux discontinuities, while the macro-scale model needs to only model thermal conduction using continuous (homogenized) fields. The convergence behavior of the tightly coupled macro-micro finite element scheme with respect to decreasing element size is analyzed by comparing with a known analytical solution of the Stefan problem.

Multi‐scale modeling of moving interface problems with flux and field jumps: Application to oxidative degradation of ceramic matrix composites

“…Boundary conditions at the micro‐scale are identified by comparing Equation (2) with (4). Thus, any such boundary conditions derived must satisfy the ‘linking condition’: A variety of boundary conditions may be derived that satisfy this constraint, however, following our previous work in 15, we focus on two possible boundary conditions (also called ‘macro–micro linking assumption’) as given below: Taylor boundary condition involves specification of density field at all points in the microstructure. The approach is called ‘Taylor model’ based on similar terminology used in multi‐scale deformation problems where displacements are fully specified at micro‐scale.…”

“…In the above equation, gradient in partial density of specie i at the macroscopic material point is denoted as . The most general assumption behind homogenization theory is that the gradient as seen at the macro‐scale can be represented purely in terms of the field variables at the exterior boundary of the microstructure 15, 17: Using the micro‐scale field decomposition (Equation (1)), it can be shown that: We employ the generalized divergence theorem of the form ∫ V ∇χd V = ∫ S χ n d S + ∫[|χ|] n I d S I in the above equation (where [|χ|] denotes the jump in the field quantity across the evolving interface) to obtain the following relationship: In the above equation, [|ρ|] denotes the jump in partial density of specie i across the interface ( S I ) with normal n I . The jump in field across an interface is computed as [|ρ|] = ρ + −ρ − .…”

“…This forms the basis for micro‐to‐macro linking . In particular, we are interested in obtaining a macroscopic flux that satisfies Hills macro‐homogeneity condition (which relates the macroscopic flux with its microstructural counterpart ( q 15, 19) as follows: The fluxes need to be derived such that the above macro‐homogeneity condition is satisfied when using either Taylor or homogenization boundary conditions as follows: The Taylor model. Using boundary conditions (Equations (6) and (7)) in the macro‐homogeneity condition (Equation (14)) leads to the following equation: Comparing Equation (15) with (14): The Homogenization model.…”

“…The reference density ρ ref is obtained using the macro–micro balance of mass condition and Equation (1). This definition is consistent with the condition that stored mass at macro‐scale is same as the average micro‐scale stored mass 15. To further understand the micro‐scale quantities that are needed to create the overall system of equations, the Jacobian matrix and force vector for a finite element e with shape functions N i occupying a volume Ω e are expanded as follows: From the above equations, it is seen that homogenized diffusivities relating the mass flux of species A with respect to the pressure gradient of species B need to be defined at each integration point in the macro‐scale.…”

“…In solidification problems, a flux jump (manifested as latent heat) occurs at the evolving solidification front whereas the overall temperature field itself is continuous in the domain. Recently, the homogenization scheme was extended for addressing such problems in our recent work 15. Oxidation problem addressed in this work not only involves a flux discontinuity but also additionally involves a field discontinuity in the form of a jump in oxygen density field across the oxidizing interface.…”