Generalized Multiscale Finite Element Methods for a Time-Dependent Free-Boundary Model of Resin Transfer Molding

We study the resin injection stage in resin transfer molding (RTM), where a liquid polymer advances through a heterogeneous porous preform and creates a moving saturated/unsaturated interface. The process is formulated as a time-dependent free-boundary problem for pressure and saturation, with the phase constraint represented by a multivalued Heaviside graph. A time-discrete variational-inequality formulation is recast through a duality method based on Yosida approximations, leading to a nonlinear iteration for advancing the filling front. To treat high-contrast permeability fields without assuming scale separation, we embed the Generalized Multiscale Finite Element Method (GMsFEM) in this nonlinear procedure and construct a reduced coarse-scale pressure solver enriched for nonhomogeneous inlet data and internal low-permeability inclusions. The contribution is the RTM-focused integration of established free-boundary, Yosida-duality, and multiscale techniques into a filling algorithm for heterogeneous porous preforms. Numerical experiments with rectangular and irregular inclusions show that the multiscale approximation reproduces the main pressure and saturation patterns of the fine-grid reference solution on an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>8</m:mn> <m:mo>×</m:mo> <m:mn>8</m:mn> </m:mrow> </m:math> {8× 8} coarse grid with retained local basis functions.